Explicit Solution of the Inverse Eigenvalue Problem of Real Symmetric Matrices and Its Application to Electrical Network Synthesis

Abstract

A novel procedure for explicit construction of the entries of real symmetric matrices with assigned spectrum and the entries of the corresponding orthogonal modal matrices is presented. The inverse eigenvalue problem of symmetric matrices with some specific sign patterns (including hyperdominant one) is explicitly solved too. It has been shown to arise thereof a possibility of straightforward solving the inverse eigenvalue problem of symmetric hyperdominant matrices with assigned nonnegative spectrum. The results obtained are applied thereafter in synthesis of driving-point immittance functions of transformerless, common-ground, two-element-kind RLC networks and in generation of their equivalent realizations.

1. Introduction

During the past few decades, many papers [1–16] have studied the inverse eigenvalue problems (IEPs) of various types. The solution existence of the specific IEPs was generally considered in [1, 3–8, 10, 11, 13, 14] without explicit formulation of the corresponding procedure for solution construction, whereas in [2, 9, 12, 15, 16] this has been accomplished. The main result of [16] is the proof that IEP of symmetric hyperdominant (hd) matrices with assigned nonnegative spectrum has at least one solution which has also been constructed. This settled an old IEP opened in [17]. Hyperdominant matrices have nonnegative diagonal and nonpositive off-diagonal entries and nonnegative hd margins of rows (hd margin of a row is the sum of entries in that row). The tool used in [16] to construct the 𝑛th-order hd matrix with assigned spectrum was the 𝑛th-order orthogonal Hessenberg matrix constructed as a special product of 𝑛−1 plane rotations [15]. Hessenberg matrices naturally arise in study of symmetric tridiagonal matrices, skew symmetric, and orthogonal matrices [13, 14, 18]. A matrix is upper (lower) Hessenberg if its entry (𝑘, 𝑚) vanishes whenever 𝑘>𝑚+1(𝑚>𝑘+1).

In practical work, it is commonly assumed to be better not to form Hessenberg matrices explicitly, but to keep them as products of plane rotations. On the other hand, explicit construction of real symmetric matrices with nonnegative spectrum, which either have hd sign pattern or are truly hd, is proved to be an inevitable task in considering the synthesis of driving-point immittance functions of passive, transformerless, common-ground, two element-kind 𝑅𝐿𝐶 networks and in generation of their equivalent realizations [17–19]. 𝑅𝐿𝐶 networks are comprised solely of resistors (𝑅), inductors (𝐿), and capacitors (𝐶). Driving-point immittance function of a lumped, time invariant, linear electrical network is either a driving-point impedance 𝑍(𝑠), or a driving-point admittance 𝑌(𝑠)=𝑍⁻¹(𝑠) (𝑠=𝜎+𝑗⋅𝜔 is the complex frequency; 𝜎, 𝜔 are real numbers; 𝑗∶=√−1). It is well known that a real rational function in 𝑠 can be driving-point immittance function of 𝑅𝐿𝐶 network if and only if it is positive real function in 𝑠; or similarly, a necessary condition for a stable square matrix 𝐖(𝑠) of real rational functions in 𝑠 to be driving-point immittance matrix of a passive 𝑅𝐿𝐶 network is that 𝐖(𝑠) be positive real matrix [20, 21]. A few tests for ascertaining positive real properties of functions and/or matrices can be found in [20, 21]. In [22] it has been pointed out the role of hd matrices in synthesis of both passive and active, transformerless, common-ground multiports. Unlike [16], this paper presents explicit construction of entries of real symmetric matrices with arbitrarily assigned spectrum and the entries of the corresponding orthogonal modal matrices. It also presents explicit construction of real symmetric matrices with assigned spectrum and with specific sign patterns (including hd one). Thereof, a solution to the IEP of symmetric, truly hd matrices with assigned nonnegative spectrum is produced. Some of the obtained results are then applied in synthesis of driving-point immitances of transformerless, common-ground, two-element-kind 𝑅𝐿𝐶 networks and in generation of their equivalent realizations. The two proposed realization procedures are illustrated by an example. Throughout the paper ⊕ denotes direct sum, 𝐱^𝑇 denotes transpose of 𝐱, bold capital letters denote matrices and 𝐈_𝑘 stands for the 𝑘th-order unit or identity matrix.

2. Explicit Solution to the IEP of Real Symmetric Matrices by Using Canonic Orthogonal Transformations

Let {𝜆₁,𝜆₂,…,𝜆_𝑛} be assigned spectrum of the sought real symmetric matrices and let 𝐆₁∶=diag(𝜆₁,𝜆₂,…,𝜆_𝑛) be 𝑛×𝑛 spectral matrix. Consider a set of 2×2 orthogonal matrices 𝐏_𝑘∈{𝐀_𝑘,𝐁_𝑘,𝐂_𝑘,𝐃_𝑘}(𝑘=1,…,𝑛−1): 𝐏_𝑘∶=𝑎^𝑘𝑏_𝑘𝑐_𝑘𝑑_𝑘,𝐀^𝑘∶=cos𝜃^𝑘−sin𝜃_𝑘sin𝜃_𝑘cos𝜃_𝑘,𝐁^𝑘∶=cos𝜃^𝑘sin𝜃_𝑘𝐂_𝑘sin𝜃_𝑘−cos𝜃_𝑘,∶=cos𝜃^𝑘sin𝜃_𝑘−sin𝜃_𝑘cos𝜃_𝑘,𝐃^𝑘∶=−cos𝜃^𝑘sin𝜃_𝑘sin𝜃_𝑘cos𝜃_𝑘,𝜃^𝑘∈0,𝜋2,(2.1)which are either rotators (𝐀_𝑘 and 𝐂_𝑘) or reflectors (𝐁_𝑘 and 𝐃_𝑘). A useful set of 𝑛×𝑛 orthogonal matrices is 𝐔₁∶=𝐏₁⊕𝐈_𝑛−2,𝐔_𝑘∶=𝐈_𝑘−1⊕𝐏_𝑘⊕𝐈_{𝑛−𝑘−1}(𝑘=2,…,𝑛−2),𝐔_𝑛−1∶=𝐈_𝑛−2⊕𝐏_𝑛−1.(2.2) From the following two matrix recurrent relations 𝐆_𝑘+1∶=𝐔_𝑘𝐆_𝑘𝐔^T_𝑘,𝐒_𝑘+1∶=𝐔_𝑛−𝑘𝐒_𝑘𝐔^T_𝑛−𝑘(𝑘=1,…,𝑛−1),(2.3) we readily obtain 𝑛×𝑛 real symmetric matrices 𝐆_𝑛 and 𝐒_𝑛, which are both congruent and similar to 𝐆₁𝐆_𝑛=𝐔𝐆₁𝐔^T,𝐔∶=𝐔_𝑛−1𝐔_𝑛−2⋯𝐔₁;𝐒_𝑛=𝐕𝐆₁𝐕^T,𝐕∶=𝐕₁𝐕₂⋯𝐕_𝑛−1.(2.4) Columns of the orthogonal modal matrix 𝐔(𝐕) correspond to eigenvectors of 𝐆_𝑛(𝐒_𝑛). Out of (𝑛−1)! different possibilities of using (2.3) in generation of 𝐆_𝑛 and 𝐒_𝑛, only the two selected by (2.4) produce explicit expressions of entries of 𝐆_𝑛 and 𝐒_𝑛 in terms of {𝜆₁,𝜆₂,…,𝜆_𝑛} and the entries of 𝐏_𝑘(𝑘=1,…,𝑛−1). 𝐔(𝐕) from (2.4) will be shown later to take on lower (upper) Hessenberg form with the entries explicitly expressed too. For the sake of brevity, we will restrict our consideration only to the first of relations (2.3), bearing on mind the possibility of treating the second one similarly. For 𝑘=1 and 𝑘=2 we readily obtain 𝐆₂ and 𝐆₃, by using (2.1) and (2.3): 𝐆₂==⎡⎢⎢⎢⎣𝐏𝑎₁₁⊕𝐈_𝑛−2𝐆₁𝐏^T₁⊕𝐈_𝑛−2𝑏₁𝑐₁𝑑₁𝟎_2,𝑛−2𝟎^T_2,𝑛−2𝐈_𝑛−2⎤⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜆¹00𝜆₂𝟎_2,𝑛−2𝟎^T_2,𝑛−2𝜆₃𝟎⋱𝟎𝜆_𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎣𝑎₁𝑐₁𝑏₁𝑑₁𝟎_2,𝑛−2𝟎^T_2,𝑛−2𝐈_𝑛−2=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜆₁⎤⎥⎥⎥⎦𝑎²¹+𝜆₂𝑏²¹𝜆₁−𝜆₂𝑎₁𝑐₁𝜆₁−𝜆₂𝑎₁𝑐₁𝜆₁𝑐²¹+𝜆₂𝑑²¹𝟎_2,𝑛−2𝟎^T_2,𝑛−2𝜆₃𝟎⋱𝟎𝜆_𝑛⎧⎪⎪⎨⎪⎪⎩More𝑎generally,∀𝑘=1,…,𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦−,1itholds_𝑘𝑎𝑏_𝑘+𝑐_𝑘𝑑_𝑘=𝑎_𝑘𝑐_𝑘+𝑏_𝑘𝑑_𝑘=0_2𝑘𝐆+𝑏^2𝑘=𝑐^2𝑘+𝑑^2𝑘=𝑎^2𝑘+𝑐^2𝑘=𝑏^2𝑘+𝑑^2𝑘=1₃⎫⎪⎪⎬⎪⎪⎭,==⎡⎢⎢⎢⎢⎢⎣110⊕𝑎0𝐏₂₂⊕0𝐈_𝑛−3𝐆₂1⊕𝐏^T₂⊕𝐈_𝑛−3𝑏₂0𝑐₂𝑑₂𝟎_3,𝑛−3𝟎^T_3,𝑛−3𝐈_𝑛−3⎤⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜆¹𝑎²¹+𝜆₂𝑏²¹𝜆₁−𝜆₂𝑎₁𝑐₁𝜆₁−𝜆₂𝑎₁𝑐₁𝜆₁𝑐²¹+𝜆₂𝑑²¹𝟎_2,𝑛−2𝟎^T_2,𝑛−2𝜆₃𝟎⋱𝟎𝜆_𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎣10𝑎0₂0𝑐₂0𝑏₂𝑑₂𝟎_3,𝑛−3𝟎^T_3,𝑛−3𝐈_𝑛−3=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜆₁⎤⎥⎥⎥⎥⎥⎦𝑎²¹+𝜆₂𝑏²¹(𝜆₁−𝜆₂)𝑎₁𝑎₂𝑐₁(𝜆₁−𝜆₂)𝑎₁𝑐₁𝑐₂(𝜆₁−𝜆₂)𝑎₁𝑎₂𝑐₁(𝜆₁𝑐²¹+𝜆₂𝑑²¹)𝑎²²+𝜆₃𝑏²²(𝜆₁𝑐²¹+𝜆₂𝑑²¹−𝜆₃)𝑎₂𝑐₂(𝜆₁−𝜆₂)𝑎₁𝑐₁𝑐₂(𝜆₁𝑐²¹+𝜆₂𝑑²¹−𝜆₃)𝑎₂𝑐₂(𝜆₁𝑐²¹+𝜆₂𝑑²¹)𝑐²²+𝜆₃𝑑²²𝟎_3,𝑛−3𝟎^T_3,𝑛−3𝜆₄𝟎⋱𝟎𝜆_𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(2.5) Let 𝜆^∗1∶=𝜆₁, 𝜀₁∶=(𝜆^∗1−𝜆₂)𝑎₁, and 𝐱₂∶=𝑐₁𝜀₁, and let us firstly introduce in (2.5) the following notation: 𝐌₂∶=𝜆₁𝑎𝐀²¹₂+𝜆₂𝑏²¹,𝜆^∗2∶=𝜆^∗1𝑐²¹+𝜆₂𝑑²¹,𝐃^∗2∶=diag𝜆^∗2,𝜆₃,𝐃₂∶=diag𝜆₄,…,𝜆_𝑛,∶=[𝐱₂∣0]^T,𝐌₃∶=𝜆¹𝑎²¹+𝜆₂𝑏²¹𝜆₁−𝜆₂𝑎₁𝑎₂𝑐₁𝜆_∗3𝜆₁−𝜆₂𝑎₁𝑎₂𝑐₁𝜆₁𝑐²¹+𝜆₂𝑑²¹𝑎²²+𝜆₃𝑏²²,𝐃₃∶=𝜆₁𝑐²¹+𝜆₂𝑑²¹𝑐²²+𝜆₃𝑑²²,𝐃^∗3∶=diag𝜆^∗3,𝜆₄,∶=diag𝜆₅,…,𝜆_𝑛,𝐀₃∶=𝜆¹−𝜆₂𝑎₁𝑐₁𝑐₂𝜆₁𝑐²¹+𝜆₂𝑑²¹−𝜆₃𝑎₂𝑐₂00.(2.6) Thereafter, observing the partition of 𝐆₂ and 𝐆₃ obtained in (2.5) 𝐆₂=⎡⎢⎢⎢⎣𝐌²𝐀^T₂𝟎_1,𝑛−3𝐀₂𝐃^∗2𝟎_2,𝑛−3𝟎^T_1,𝑛−3𝟎^T_2,𝑛−3𝐃₂⎤⎥⎥⎥⎦,𝐆₃=⎡⎢⎢⎢⎣𝐌³𝐀^T₃𝟎_2,𝑛−4𝐀₃𝐃^∗3𝟎_2,𝑛−4𝟎^T_2,𝑛−4𝟎^T_2,𝑛−4𝐃₃⎤⎥⎥⎥⎦,(2.7) it can readily be anticipated the partition of subsequent matrices 𝐆_𝑘(𝑘=4,…,𝑛−2) as follows: 𝐆_𝑘=⎡⎢⎢⎢⎣𝐌^𝑘𝐀^T_𝑘𝟎_{𝑘−1,𝑛−𝑘−1}𝐀_𝑘𝐃^∗𝑘𝟎_{2,𝑛−𝑘−1}𝟎^T_{𝑘−1,𝑛−𝑘−1}𝟎^T_{2,𝑛−𝑘−1}𝐃_𝑘⎤⎥⎥⎥⎦,𝐱_𝑘∶=𝑥_𝑘,1𝑥_𝑘,2⋯𝑥_{𝑘,𝑘−1},𝐀_𝑘∶=𝐱^𝑘𝟎_1,𝑘−1,(2.8) where 𝐌_𝑘 is the symmetric (𝑘−1)×(𝑘−1) matrix, 𝐱_𝑘 is 1×(𝑘−1) row vector, 𝐀_𝑘 is 2×(𝑘−1) matrix, 𝜆^∗𝑘 is modified eigenvalue 𝜆_𝑘, 𝐃^∗𝑘∶=diag(𝜆^∗𝑘,𝜆_𝑘+1) and 𝐃_𝑘∶=diag(𝜆_𝑘+2,…,𝜆_𝑛). For 𝑘=2,…,𝑛−3 from (2.1)–(2.3), (2.8) it follows that 𝐆_𝑘+1=𝐈_𝑘−1⊕𝐏_𝑘⊕𝐈_{𝑛−𝑘−1}𝐆_𝑘𝐈_𝑘−1⊕𝐏^T_𝑘⊕𝐈_{𝑛−𝑘−1}=⎡⎢⎢⎢⎢⎣𝐌^𝑘𝐀^T_𝑘𝐏^T_𝑘𝟎_{𝑘−1,𝑛−𝑘−1}𝐏_𝑘𝐀_𝑘𝐏_𝑘𝐃^∗𝑘𝐏^T_𝑘𝟎_{2,𝑛−𝑘−1}𝟎^T_{𝑘−1,𝑛−𝑘−1}𝟎^T_{2,𝑛−𝑘−1}𝐃_𝑘𝐏_𝑘⎤⎥⎥⎥⎥⎦,𝐀_𝑘=𝑎^𝑘𝑥_𝑘,1⋯𝑎_𝑘𝑥_{𝑘,𝑘−1}𝑐_𝑘𝑥_𝑘,1⋯𝑐_𝑘𝑥_{𝑘,𝑘−1},𝐏^𝑘𝐃^∗𝑘𝐏^T_𝑘=𝜆^∗𝑘𝑎^2𝑘+𝜆_𝑘+1𝑏^2𝑘𝜆^∗𝑘−𝜆_𝑘+1𝑎_𝑘𝑐_𝑘𝜆^∗𝑘−𝜆_𝑘+1𝑎_𝑘𝑐_𝑘𝜆^∗𝑘𝑐^2𝑘+𝜆_𝑘+1𝑑^2𝑘.(2.9) For 𝑘=2,…,𝑛−3, let us define: 𝜆^∗𝑘+1∶=𝜆^∗𝑘𝑐^2𝑘+𝜆_𝑘+1𝑑^2𝑘, 𝜀_𝑘∶=(𝜆^∗𝑘−𝜆_𝑘+1)𝑎_𝑘, 𝜓_𝑘𝑘∶=𝜆^∗𝑘𝑎^2𝑘+𝜆_𝑘+1𝑏^2𝑘 and thereafter 𝐃_𝑘+1∶=diag(𝜆_𝑘+3,…,𝜆_𝑛) and 𝐃^∗𝑘+1∶=diag(𝜆^∗𝑘𝑐^2𝑘+𝜆_𝑘+1𝑑^2𝑘,𝜆_𝑘+2). Then, from (2.8)-(2.9) it follows the identification 𝐌_𝑘+1∶=𝐌^𝑘𝑎_𝑘𝐱^T_𝑘𝑎_𝑘𝐱_𝑘𝜓_𝑘𝑘,𝐀^𝑘+1=𝐱^𝑘+1𝟎_1,𝑘∶=𝑐^𝑘𝐱_𝑘𝑐_𝑘𝜀_𝑘𝟎_1,𝑘0=𝑐^𝑘𝑥_𝑘,1⋯𝑐_𝑘𝑥_{𝑘,𝑘−1}𝑐_𝑘𝜀_𝑘0⋯00,(2.10) which enables the partition of (2.10)𝐆_𝑘+1 in () to be like that of (2.10)𝐆_𝑘 in (), and that partition of (2.10)𝐱_𝑘+1 be rather simple 𝐆_𝑘+1=⎡⎢⎢⎢⎣𝐌^𝑘+1𝐀^T_𝑘+1𝟎_{𝑘,𝑛−𝑘−2}𝐀_𝑘+1𝐃^∗𝑘+1𝟎_{2,𝑛−𝑘−2}𝟎^T_{𝑘,𝑛−𝑘−2}𝟎^T_{2,𝑛−𝑘−2}𝐃_𝑘+1⎤⎥⎥⎥⎦,𝐱_𝑘+1∶=𝑐_𝑘𝐱^𝑘∣𝜀_𝑘,𝑘=2,…,𝑛−3.(2.10) Let 𝜓₁₁∶=𝐌₂. Having uncovered the partition pattern of 𝐌_𝑘+1(𝑘=2,…,𝑛−3), we can pursue partitioning of 𝐌_𝑛−2 backwardly from 𝐌_𝑛−2 to 𝐌₂, by using (2.10)). Afterwards, we can produce 𝐆_𝑛−2, by using (2.10)-(2.11). The results are 𝐌_𝑛−2=⎡⎢⎢⎢⎢⎢⎣𝜓¹¹𝑎₂𝐱^T₂𝑎₂𝐱₂𝜓₂₂𝑎₃𝐱^T₃𝑎₃𝐱₃𝜓₃₃⋮⋱⋯𝑎_𝑛−3𝐱^T_𝑛−3𝑎_𝑛−3𝐱_𝑛−3𝜓_{𝑛−3,𝑛−3}⎤⎥⎥⎥⎥⎥⎦,𝐆_𝑛−2=⎡⎢⎢⎢⎣𝐌^𝑛−2𝐱^T_𝑛−2𝐱_𝑛−2𝜆^∗𝑛−2𝟎_𝑛−2,1𝟎_1,𝑛−2𝜆_𝑛−1𝟎_𝑛−1,1𝟎_1,𝑛−1𝜆_𝑛⎤⎥⎥⎥⎦.(2.12) Since 𝐆_𝑛−1∶=(𝐈_𝑛−3⊕𝐏_𝑛−2⊕1)𝐆_𝑛−2(𝐈_𝑛−3⊕𝐏^T_𝑛−2⊕1) and 𝐏_𝑛−2𝜆^∗𝑛−20𝜆0_𝑛−1𝐏^T𝑛−2=𝜆^∗𝑛−2𝑎^2𝑛−2+𝜆_𝑛−1𝑏^2𝑛−2𝜆^∗𝑛−2−𝜆_𝑛−1𝑎_𝑛−2𝑐_𝑛−2𝜆^∗𝑛−2−𝜆_𝑛−1𝑎_𝑛−2𝑐_𝑛−2𝜆^∗𝑛−2𝑐^2𝑛−2+𝜆_𝑛−1𝑑^2𝑛−2,(2.13) then after defining 𝜓_{𝑛−2,𝑛−2}∶=𝜆^∗𝑛−2𝑎^2𝑛−2+𝜆_𝑛−1𝑏^2𝑛−2,𝜀_𝑛−2∶=(𝜆^∗𝑛−2−𝜆_𝑛−1)𝑎_𝑛−2,𝜆^∗𝑛−1∶=𝜆^∗𝑛−2𝑐^2𝑛−2+𝜆_𝑛−1𝑑^2𝑛−2 and 𝐱_𝑛−1∶=𝑐_𝑛−2[𝐱_𝑛−2∣𝜀_𝑛−2], it follows from (2.12)-(2.13) 𝐆_𝑛−1=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜓¹¹𝑎₂𝐱^T₂𝑎₂𝐱₂𝜓₂₂𝑎₃𝐱^T₃𝑎₃𝐱₃𝜓₃₃⋮⋱⋯𝑎_𝑛−3𝐱^T_𝑛−3𝑎_𝑛−3𝐱_𝑛−3𝜓_{𝑛−3,𝑛−3}𝑎_𝑛−2𝐱^T_𝑛−2𝑎_𝑛−2𝐱_𝑛−2𝜓_{𝑛−2,𝑛−2}𝐱^T_𝑛−1𝐱_𝑛−1𝜆^∗𝑛−1𝟎_𝑛−1,1𝟎_1,𝑛−1𝜆_𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(2.14) Since 𝐆_𝑛∶=(𝐈_𝑛−2⊕𝐏_𝑛−1)𝐆_𝑛−1(𝐈_𝑛−2⊕𝐏^T_𝑛−1) and 𝐏_𝑛−1𝜆^∗𝑛−10𝜆0_𝑛𝐏^T𝑛−1=𝜆^∗𝑛−1𝑎^2𝑛−1+𝜆_𝑛𝑏^2𝑛−1𝜆^∗𝑛−1−𝜆_𝑛𝑎_𝑛−1𝑐_𝑛−1𝜆^∗𝑛−1−𝜆_𝑛𝑎_𝑛−1𝑐_𝑛−1𝜆^∗𝑛−1𝑐^2𝑛−1+𝜆_𝑛𝑑^2𝑛−1,(2.15) then on introducing 𝜓_{𝑛−1,𝑛−1}∶=𝜆^∗𝑛−1𝑎^2𝑛−1+𝜆_𝑛𝑏^2𝑛−1,𝜀_𝑛−1∶=(𝜆^∗𝑛−1−𝜆_𝑛)𝑎_𝑛−1, and 𝜆^∗𝑛∶=𝜆^∗𝑛−1𝑐^2𝑛−1+𝜆_𝑛𝑑^2𝑛−1, we obtain from (2.14)-(2.15) the partition of 𝐆_𝑛 which is amenable to the production of its entries in explicit form and is suitable for further discussion about solving some specific IEPs 𝐆_𝑛=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜓¹¹𝑎₂𝐱^T₂𝑎₂𝐱₂𝜓₂₂𝑎₃𝐱^T₃𝑎₃𝐱₃𝜓₃₃⋮⋱⋯𝑎_𝑛−2𝐱^T_𝑛−2𝑎_𝑛−2𝐱_𝑛−2𝜓_{𝑛−2,𝑛−2}𝑎_𝑛−1𝐱^T_𝑛−1𝑐_𝑛−1𝐱^T_𝑛−1𝑎_𝑛−1𝐱_𝑛−1𝜓_{𝑛−1,𝑛−1}𝑐_𝑛−1𝜀_𝑛−1𝑐_𝑛−1𝐱_𝑛−1𝑐_𝑛−1𝜀_𝑛−1𝜆^∗𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦.(2.16) For 𝑘=2,…,𝑛, we consecutively obtain from 𝜆^∗1∶=𝜆₁ and 𝜆^∗𝑘∶=𝜆^∗𝑘−1𝑐^2𝑘−1+𝜆_𝑘𝑑^2𝑘−1 that generally it holds 𝜆^∗𝑘=𝑐₁𝑐₂⋯𝑐_𝑘−1²𝜆₁+𝑑₁𝑐₂⋯𝑐_𝑘−1²𝜆₂+⋯+𝑑_𝑘−2𝑐_𝑘−1²𝜆_𝑘−1+𝑑^2𝑘−1𝜆_𝑘,𝑘=2,…,𝑛.(2.17) Since (2.17)𝜓₁₁=𝜆₁𝑎²¹+𝜆₂𝑏²¹ and (2.17)𝜓_𝑘𝑘∶=𝜆^∗𝑘𝑎^2𝑘+𝜆_𝑘+1𝑏^2𝑘(𝑘=2,…,𝑛−1), then from () it follows that 𝜓_𝑘𝑘=𝑐₁𝑐₂⋯𝑐_𝑘−1𝑎_𝑘²𝜆₁+𝑑₁𝑐₂⋯𝑐_𝑘−1𝑎_𝑘²𝜆₂+⋯+𝑑_𝑘−2𝑐_𝑘−1𝑎_𝑘²𝜆_𝑘−1+𝑑_𝑘−1𝑎_𝑘²𝜆_𝑘+𝑏^2𝑘𝜆_𝑘+1,𝑘=2,…,𝑛−1.(2.17) Observe that it is not necessary to calculate “𝜓”s from (2.18), but only the modified eigenvalues from (2.17) since it holds 𝜀_𝑘=(𝜆^∗𝑘−𝜆_𝑘+1)𝑎_𝑘 and 𝜓_𝑘𝑘=𝜆^∗𝑘𝑎^2𝑘+𝜆_𝑘+1𝑏^2𝑘=(𝜆^∗𝑘−𝜆_𝑘+1)𝑎^2𝑘+(𝑎^2𝑘+𝑏^2𝑘)𝜆_𝑘+1=𝑎_𝑘𝜀_𝑘+𝜆_𝑘+1(𝑘=1,…,𝑛−1). As it is 𝐱₂=𝑐₁𝜀₁, then for 𝑘=2,…,𝑛−2 from (2.10)) it follows that 𝐱_𝑘+1==𝑐𝑐_𝑘𝑘𝐱^𝑘∣𝜀_𝑘=𝑐_𝑘𝑐_𝑘−1𝐱^𝑘−1∣𝜀_𝑘−1∣𝜀_𝑘=⋯=𝑐^𝑘−1𝑐𝐱_𝑘−1𝑘∣𝑐_𝑘𝑐_𝑘−1𝜀_𝑘−1∣𝑐_𝑘𝜀_𝑘=𝑐_𝑘𝑐_𝑘−1⋯𝑐₂𝐱₂∣𝑐_𝑘𝑐_𝑘−1⋯𝑐₂𝜀₂∣⋯∣𝑐_𝑘𝑐_𝑘−1𝜀_𝑘−1∣𝑐_𝑘𝜀_𝑘𝑎𝑐_𝑘−1⋯𝑐₂𝑐₁𝜀₁∣𝑐_𝑘𝑐_𝑘−1⋯𝑐₂𝜀₂∣⋯∣𝑐_𝑘𝑐_𝑘−1𝜀_𝑘−1∣𝑐_𝑘𝜀_𝑘(2.19)_𝑘𝐱_𝑘=𝑎_𝑘𝑐_𝑘−1𝑐_𝑘−2⋯𝑐₂𝑐₁𝜀₁∣𝑎_𝑘𝑐𝑘^𝑘−1𝑐=2^𝑘−2,⋯…𝑐,²𝜀𝑛²−∣1.⋯∣𝑎_𝑘𝑐_𝑘−1𝑐_𝑘−2𝜀_𝑘−2∣𝑎_𝑘𝑐_𝑘−1𝜀_𝑘−1,(2.20) The real symmetric matrix 𝐆_𝑛 with assigned spectrum {𝜆₁,𝜆₂,…,𝜆_𝑛} and the explicitly expressed entries can be derived from (2.16) and (2.20), bearing on mind that “𝜓”s and “𝜀”s are calculated by using {𝜆₁,𝜆₂,…,𝜆_𝑛}, 𝐏_𝑘, modified eigenvalues (2.17) and 𝜀_𝑘=(𝜆^∗𝑘−𝜆_𝑘+1)𝑎_𝑘(𝑘=1,…,𝑛−1): 𝐆_𝑛=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜓¹¹𝑎₂𝑎₂𝑐₁𝜀₁𝑎₃𝑐₂𝑐₁𝜀₁𝑎₄𝑐₃𝑐₂𝑐₁𝜀₁⋯𝒫𝒮𝑎₃𝑐₁𝜀₁𝜓₂₂𝑎₃𝑐₂𝜀₂𝑎₄𝑐₃𝑐₂𝜀₂⋯𝒰ℱ𝑎₄𝑐₂𝑐₁𝜀₁𝑎₃𝑐₂𝜀₂⋮⋮⋯⋮⋮𝑐₃⋮𝑐₂𝑐₁𝜀₁𝑎₄𝑐₃𝑐₂𝜀₂⋮⋮⋮⋮⋮𝒫⋮⋮⋮⋱⋮⋮𝒰⋯⋯⋯𝜓_{𝑛−1,𝑛−1}𝑐_𝑛−1𝜀_𝑛−1𝒮ℱ⋯⋯⋯𝑐_𝑛−1𝜀_𝑛−1𝜆^∗𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(2.21) where 𝒫 denotes 𝑎_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑐₁𝜀₁, 𝒮 denotes 𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑐₁𝜀₁, 𝒰 denotes 𝑎_𝑛−1𝑐_𝑛−2⋯𝑐₂𝜀₂, and 𝐹 denotes 𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₂𝜀₂. Entries of 𝐆_𝑛=𝐆^T_𝑛=[𝑔_𝑘𝑚]_𝑛×𝑛 are 𝑔_𝑘𝑘=𝜓_𝑘𝑘(𝑘=1,…,𝑛−1), 𝑔_𝑛𝑛=𝜆^∗𝑛, 𝑔_𝑘𝑚=𝑎_𝑘𝑐_𝑘−1𝑐_𝑘−2⋯𝑐_𝑚𝜀_𝑚(𝑘>𝑚;𝑘=2,…,𝑛−1) and 𝑔_𝑛𝑚=𝑐_𝑛−1𝑐_𝑛−2⋯𝑐_𝑚𝜀_𝑚(𝑚=1,…,𝑛−1). They are calculated according to the following steps:

(a)Select arbitrarily the entries {𝑎_𝑘,𝑏_𝑘,𝑐_𝑘,𝑑_𝑘} of 2×2 orthogonal matrices 𝐏_𝑘(𝑘=1,…,𝑛−1), given by (2.1);(b)with 𝜆^∗1∶=𝜆₁, calculate the modified eigenvalues 𝜆^∗𝑘(𝑘=2,…,𝑛), by using (2.17);(c)calculate 𝜀_𝑘=(𝜆^∗𝑘−𝜆_𝑘+1)𝑎_𝑘 and 𝜓_𝑘𝑘=𝑎_𝑘𝜀_𝑘+𝜆_𝑘+1(𝑘=1,…,𝑛−1);(d)calculate the entries of 𝐆_𝑛, by using (2.21).

Matrix 𝐔 (2.4) is 𝑛×𝑛 orthogonal modal matrix established from eigenvectors of 𝐆_𝑛. We will now prove that 𝐔 is not only orthogonal, but also lower Hessenberg with explicitly expressed entries. Let us firstly produce 𝐔^T₁𝐔^T₂ and 𝐔^T₁𝐔^T₂𝐔^T₃, whose partition will enable us to anticipate the partition of 𝐔^T₁𝐔^T₂𝐔^T₃⋯𝐔^T_𝑘(𝑘=4,…,𝑛−1)𝐔^T₁𝐔^T₂=⎡⎢⎢⎢⎣𝑎¹𝑎₂𝑐₁𝑐₂𝑐₁𝑏₁𝑎₂𝑑₁𝑐₂𝑑₁0𝑏₂𝑑₂𝟎_3,𝑛−3𝟎^T_3,𝑛−3𝐈_𝑛−3⎤⎥⎥⎥⎦,𝐔_T1𝐔^T₂𝐔^T₃=⎡⎢⎢⎢⎢⎢⎣𝑎¹𝑎₂𝑐₁𝑎₃𝑐₂𝑐₁𝑐₃𝑐₂𝑐₁𝑏₁𝑎₂𝑑₁𝑎₃𝑐₂𝑑₁𝑐₃𝑐₂𝑑₁0𝑏₂𝑎₃𝑑₂𝑐₃𝑑₂00𝑏₃𝑑₃𝟎_4,𝑛−4𝟎^T_4,𝑛−4𝐈_𝑛−4⎤⎥⎥⎥⎥⎥⎦.(2.22) If we now suppose that 𝐔^T₁𝐔^T₂⋯𝐔^T_𝑘∶=𝐇_𝑘+1⊕𝐈_{𝑛−𝑘−1}(𝑘=2,…,𝑛−1) where 𝐇_𝑘+1 is orthogonal (𝑘+1)×(𝑘+1) upper Hessenberg matrix 𝐇_𝑘+1=⎡⎢⎢⎢⎢⎢⎢⎣𝑎¹𝑎₂𝑐₁𝑎₃𝑐₂𝑐₁⋯𝑎_𝑘𝑐_𝑘−1⋯𝑐₂𝑐₁𝑐_𝑘𝑐_𝑘−1⋯𝑐₂𝑐₁𝑏₁𝑎₂𝑑₁𝑎₃𝑐₂𝑑₁⋯𝑎_𝑘𝑐_𝑘−1⋯𝑐₂𝑑₁𝑐_𝑘𝑐_𝑘−1⋯𝑐₂𝑑₁0𝑏₂𝑎₃𝑑₂⋯𝑎_𝑘𝑐_𝑘−1⋯𝑐₃𝑑₂𝑐_𝑘𝑐_𝑘−1⋯𝑐₃𝑑₂⋯0⋯⋯⋯⋯⋯⋯0𝑏_𝑘−1𝑎_𝑘𝑑_𝑘−1𝑐_𝑘𝑑_𝑘−10⋯⋯0𝑏_𝑘𝑑_𝑘⎤⎥⎥⎥⎥⎥⎥⎦,(2.23) then since according to (2.2), it holds 𝐔_𝑘+1∶=𝐈_𝑘⊕𝐏_𝑘+1⊕𝐈_{𝑛−𝑘−2}, we may write further for 𝑘=2,…,𝑛−2𝐔^T₁𝐔^T₂⋯𝐔^T_𝑘𝐔^T_𝑘+1=𝐇_𝑘+1⊕𝐈_{𝑛−𝑘−1}𝐔^T_𝑘+1==𝐇𝐇_𝑘+1𝑘+1⊕1⊕𝐈_{𝑛−𝑘−2}𝐈_𝑘⊕𝐏^T_𝑘+1⊕𝐈_{𝑛−𝑘−2}⊕1𝐈_𝑘⊕𝐏^T_𝑘+1⊕𝐈_{𝑛−𝑘−2}=𝐇_𝑘+2⊕𝐈_{𝑛−𝑘−2},where𝐇_𝑘+2∶=𝐇_𝑘+1⊕1𝐈_𝑘⊕𝐏^T_𝑘+1.(2.24) By using (2.2), (2.23)-(2.24), it follows that 𝐇_𝑘+2=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑎¹𝑏₁𝑎₂𝑐₁𝑎₃𝑐₂𝑐₁⋯𝑎_𝑘𝑐_𝑘−1⋯𝑐₂𝑐₁𝑐_𝑘𝑐_𝑘−1⋯𝑐₂𝑐₁00𝑎₂𝑑₁𝑎₃𝑐₂𝑑₁⋯𝑎_𝑘𝑐_𝑘−1⋯𝑐₂𝑑₁𝑐_𝑘𝑐_𝑘−1⋯𝑐₂𝑑₁0⋯𝑏₂𝑎₃𝑑₂⋯𝑎_𝑘𝑐_𝑘−1⋯𝑐₃𝑑₂𝑐_𝑘𝑐_𝑘−1⋯𝑐₃𝑑₂00⋯⋯⋯⋯⋯⋯0⋯0𝑏_𝑘−1𝑎_𝑘𝑑_𝑘−1𝑐_𝑘𝑑_𝑘−100⋯⋯0𝑏_𝑘𝑑_𝑘0⋯⋯0001⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎣𝐈^𝑘𝟎_𝑘,2𝟎^T_𝑘,2𝑎_𝑘+1𝑐_𝑘+1𝑏_𝑘+1𝑑_𝑘+1=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑎₁⎤⎥⎥⎥⎦𝑎₂𝑐₁𝑎₃𝑐₂𝑐₁⋯𝑎_𝑘𝑐_𝑘−1⋯𝑐₂𝑐₁𝑎_𝑘+1𝑐_𝑘𝑐_𝑘−1⋯𝑐₂𝑐₁𝑐_𝑘+1𝑐_𝑘𝑐_𝑘−1⋯𝑐₂𝑐₁𝑏₁𝑎₂𝑑₁𝑎₃𝑐₂𝑑₁⋯𝑎_𝑘𝑐_𝑘−1⋯𝑐₂𝑑₁𝑎_𝑘+1𝑐_𝑘𝑐_𝑘−1⋯𝑐₂𝑑₁𝑐_𝑘+1𝑐_𝑘𝑐_𝑘−1⋯𝑐₂𝑑₁0𝑏₂𝑎₃𝑑₂⋯𝑎_𝑘𝑐_𝑘−1⋯𝑐₃𝑑₂𝑎_𝑘+1𝑐_𝑘𝑐_𝑘−1⋯𝑐₃𝑑₂𝑐_𝑘+1𝑐_𝑘𝑐_𝑘−1⋯𝑐₃𝑑₂⋯0⋯⋯⋯⋯⋯⋯⋯0𝑏_𝑘−1𝑎_𝑘𝑑_𝑘−1𝑎_𝑘+1𝑐_𝑘𝑑_𝑘−1𝑐_𝑘+1𝑐_𝑘𝑑_𝑘−10⋯⋯0𝑏_𝑘𝑎_𝑘+1𝑑_𝑘𝑐_𝑘+1𝑑_𝑘0⋯⋯00𝑏_𝑘+1𝑑_𝑘+1⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(2.25) and thereby it is proved our previous assumption that 𝐔^T₁𝐔^T₂⋯𝐔^T_𝑘∶=𝐇_𝑘+1⊕𝐈_{𝑛−𝑘−1}(𝑘=2,…,𝑛−1), where 𝐇_𝑘+1 (2.23) is the orthogonal upper Hessenberg (𝑘+1)×(𝑘+1) matrix with entries expressed explicitly. And finally, for 𝑘=𝑛−1 from 𝐔^T₁𝐔^T₂⋯𝐔^T_𝑘∶=𝐇_𝑘+1⊕𝐈_{𝑛−𝑘−1} and (2.4), (2.23), we obtain 𝐇_𝑛=𝐔^T=𝐔^T₁𝐔^T₂⋯𝐔^T_𝑛−1 and 𝐔=𝐇^T_𝑛=𝐔_𝑛−1𝐔_𝑛−2⋯𝐔₁=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑎¹𝑎₂𝑏₁00⋯00𝑎₃𝑐₁𝑎₂𝑑₁𝑏₂0⋯00𝑎₄𝑐₂𝑐₁𝑎₃𝑐₂𝑑₁𝑎₃𝑑₂⋮⋯⋮⋮𝑐⋮₃𝑐₂𝑐₁𝑎₄𝑐₃𝑐₂𝑑₁𝑎₄𝑐₃𝑑₂⋮⋮⋮⋮𝑎_𝑛−1⋮⋮⋮⋱⋮⋮𝑐_𝑛−2⋯𝑐₂𝑐₁𝑎_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑑₁⋯⋯⋯𝑎_𝑛−1𝑑_𝑛−2𝑏_𝑛−1𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑐₁𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑑₁⋯⋯⋯𝑐_𝑛−1𝑑_𝑛−2𝑑_𝑛−1⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦.(2.26) The entries of the orthogonal lower Hessenberg matrix 𝐔=[𝑢_𝑘𝑚](𝑘,𝑚=1,…,𝑛) are defined as follows: 𝑢_𝑘𝑚𝑢=_𝑘𝑘0(𝑚>𝑘+1;𝑘,𝑚=1,…,𝑛),𝑢_𝑘,𝑘+1=𝑏_𝑘(𝑘=1,…,𝑛−1),𝑢₁₁=𝑎₁,=𝑎𝑢_𝑘𝑑_𝑘−1(𝑘=2,…,𝑛−1),𝑢_𝑘,1=𝑎_𝑘𝑐_𝑘−1𝑐_𝑘−2⋯𝑐₂𝑐₁(𝑘=2,…,𝑛−1),_𝑛,1𝑢=𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₁,𝑢_𝑛,𝑘=𝑐_𝑛−1𝑐_𝑛−2⋯𝑐_𝑘𝑑_𝑘−1(𝑘=2,…,𝑛−1),_𝑘𝑚=𝑎_𝑘𝑐_𝑘−1⋯𝑐_𝑚𝑑_𝑚−1(𝑚+1≤𝑘≤𝑛−1;𝑚=2,…,𝑛−1).(2.27) By using the similar arguments as in derivation of entries of matrix 𝐔, the orthogonal matrix 𝐕 which is to be produced by using (2.4) can be shown to take on upper Hessenberg form. Proving of this fact goes with similar paces that were used for obtaining 𝐔 and it is left to the reader.

3. The Explicit Solution of the IEP of Real Symmetric Matrices with Some Specific Sign Patterns

Let the real eigenvalues from the spectrum {𝜆₁,𝜆₂,…,𝜆_𝑛} be arbitrarily enumerated, thereby establishing the sequence {𝜆_𝑘}(𝑘=1,…,𝑛). The nonnegative sequence will be denoted by {𝜆_𝑘}≥0, and the nonpositive one by {𝜆_𝑘}≤0(𝑘=1,…,𝑛). Firstly, we will prove two lemmas.

Lemma 3.1. If the sequence {𝜆_𝑘}≥0(𝑘=1,…,𝑛) is increasing [decreasing], then in (2.21) 𝜆^∗𝑛≥0,𝜓_𝑚𝑚≥0, and the sequence {𝑎_𝑚𝜀_𝑚}≤0[{𝑎_𝑚𝜀_𝑚}≥0](𝑚=1,…,𝑛−1).

Proof.Since {𝜆_𝑘}≥0(𝑘=1,…,𝑛), then it is trivial to see from (2.17) and (2.18) that all diagonal entries of 𝐆_𝑛 are nonnegative, that is, 𝜆^∗𝑛≥0 and 𝜓_𝑚𝑚≥0(𝑚=1,…,𝑛−1) no matter whether the sequence {𝜆_𝑘}≥0 is increasing or decreasing. By virtue of orthogonality of 𝐏_𝑘, we have 𝑐^2𝑘+𝑑^2𝑘=1(𝑘=1,…,𝑛). If {𝜆_𝑘}(𝑘=1,…,𝑛) is increasing sequence, then for 𝑚=1 we have 𝑎₁𝜀₁=(𝜆^∗1−𝜆₂)𝑎²¹=(𝜆₁−𝜆₂)𝑎²¹≤0 and for 𝑚=2,…,𝑛−1 we obtain 𝑑^2𝑚−1𝜆_𝑚−𝜆_𝑚+1𝑑≤_𝑚−2𝑑^2𝑚−1𝜆_𝑚−𝜆_𝑚=𝜆_𝑚1−𝑑^2𝑚−1=−𝑐^2𝑚−1𝜆_𝑚≤−𝑐^2𝑚−1𝜆_𝑚−1,𝑐_𝑚−1²𝜆_𝑚−1+𝑑^2𝑚−1𝜆_𝑚−𝜆_𝑚+1𝑑≤_𝑚−3𝑑_𝑚−2𝑐_𝑚−1²𝜆_𝑚−1−𝑐^2𝑚−1𝜆_𝑚−1=−𝑐_𝑚−2𝑐_𝑚−1²𝜆_𝑚−1≤−𝑐_𝑚−2𝑐_𝑚−1²𝜆_𝑚−2,𝑐_𝑚−2𝑐_𝑚−1²𝜆_𝑚−2+𝑑_𝑚−2𝑐_𝑚−1²𝜆_𝑚−1+𝑑^2𝑚−1𝜆_𝑚−𝜆_𝑚+1𝑑₁≤−𝑐_𝑚−3𝑐_𝑚−2𝑐_𝑚−1²𝜆_𝑚−2≤−𝑐_𝑚−3𝑐_𝑚−2𝑐_𝑚−1²𝜆_𝑚−3,𝑐₂⋯𝑐_𝑚−1²𝜆₂+⋯+𝑑_𝑚−2𝑐_𝑚−1²𝜆_𝑚−1+𝑑^2𝑚−1𝜆_𝑚−𝜆_𝑚+1≤−𝑐₁⋯𝑐_𝑚−1²𝜆₂≤−𝑐₁⋯𝑐_𝑚−1²𝜆₁.(3.1) From (2.17) and the last of inequalities (3.1) it follows 𝜆^∗𝑚≤𝜆_𝑚+1 and 𝑎_𝑚𝜀_𝑚=(𝜆^∗𝑚−𝜆_𝑚+1)𝑎^2𝑚≤0(𝑚=2,…,𝑛−1). If {𝜆_𝑘}(𝑘=1,…,𝑛) is decreasing sequence, then for 𝑚=1 we have 𝑎₁𝜀₁=(𝜆^∗1−𝜆₂)𝑎²¹=(𝜆₁−𝜆₂)𝑎²¹≥0 and for 𝑚=2,…,𝑛−1 we obtain 𝑑^2𝑚−1𝜆_𝑚−𝜆_𝑚+1𝑑≥_𝑚−2𝑑^2𝑚−1𝜆_𝑚−𝜆_𝑚=−𝜆_𝑚1−𝑑^2𝑚−1=−𝑐^2𝑚−1≥−𝑐^2𝑚−1𝜆_𝑚−1,𝑐_𝑚−1²𝜆_𝑚−1+𝑑^2𝑚−1𝜆_𝑚−𝜆_𝑚+1𝑑≥_𝑚−3𝑑_𝑚−2𝑐_𝑚−1²𝜆_𝑚−1−𝑐^2𝑚−1𝜆_𝑚−1=−𝑐_𝑚−2𝑐_𝑚−1²𝜆_𝑚−1≥−𝑐_𝑚−2𝑐_𝑚−1²𝜆_𝑚−2,𝑐_𝑚−2𝑐_𝑚−1²𝜆_𝑚−2+𝑑_𝑚−2𝑐_𝑚−1²𝜆_𝑚−1+𝑑^2𝑚−1𝜆_𝑚−𝜆_𝑚+1𝑑₁≥−𝑐_𝑚−3𝑐_𝑚−2𝑐_𝑚−1²𝜆_𝑚−2≥−𝑐_𝑚−3𝑐_𝑚−2𝑐_𝑚−1²𝜆_𝑚−3,𝑐₂⋯𝑐_𝑚−1²𝜆₂+⋯+𝑑_𝑚−2𝑐_𝑚−1²𝜆_𝑚−1+𝑑^2𝑚−1𝜆_𝑚−𝜆_𝑚+1≥−𝑐₁⋯𝑐_𝑚−1²𝜆₂≥−𝑐₁⋯𝑐_𝑚−1²𝜆₁.(3.2) From (2.17) and the last of inequalities (3.2) it follows 𝜆^∗𝑚≥𝜆_𝑚+1 and 𝑎_𝑚𝜀_𝑚=(𝜆^∗𝑚−𝜆_𝑚+1)𝑎^2𝑚≥0(𝑚=2,…,𝑛−1). This completes the proof of lemma. For a nonpositive sequence, an analogous lemma can be formulated.

Lemma 3.2. If the sequence {𝜆_𝑘}≤0(𝑘=1,…,𝑛) is increasing [decreasing], then in (2.21) 𝜆^∗𝑛≤0,𝜓_𝑚𝑚≤0 and the sequence {𝑎_𝑚𝜀_𝑚}≤0[{𝑎_𝑚𝜀_𝑚}≥0](𝑚=1,…,𝑛−1).

Proof. It is similar to that of Lemma 3.1, but in this case the diagonal entries of 𝐆_𝑛 are nonpositive, that is, 𝜆^∗𝑛≤0 and 𝜓_𝑚𝑚≤0(𝑚=1,…,𝑛−1), no matter whether the sequence {𝜆_𝑘}≤0 is increasing or decreasing (see (2.18)).

Now, we shall formulate a new theorem related to explicit solving of IEP of real symmetric matrices with some specific sign patterns.

Theorem 3.3. If 𝜃_𝑘(𝑘=1,…,𝑛−1) are arbitrarily selected angles from the range [0,𝜋/2], then the entries of real symmetric matrices 𝐆_𝑛 with assigned spectrum {𝜆₁,𝜆₂,…,𝜆_𝑛}, produced by (2.21), can attain the following twelve sign patterns (zero entries are permitted), depending on selection of matrices 𝐏_𝑘(𝑘=1,…,𝑛−1) (see (2.1)).

Case 1. 𝐏_𝑘=𝐀_𝑘=cos𝜃^𝑘−sin𝜃_𝑘sin𝜃_𝑘cos𝜃_𝑘,or𝐏^𝑘=𝐁_𝑘=cos𝜃^𝑘sin𝜃_𝑘𝐆_𝑛sin𝜃_𝑘−cos𝜃_𝑘⟹𝜆=⎡⎢⎢⎢⎢⎢⎢⎢⎣+++⋱+₁+⎤⎥⎥⎥⎥⎥⎥⎥⎦,≥𝜆₂≥⋯≥𝜆_𝑛≥0𝐆^𝑛𝜆=⎡⎢⎢⎢⎢⎢⎢⎢⎣−+−⋱+_𝑛−⎤⎥⎥⎥⎥⎥⎥⎥⎦,𝐆≤𝜆_𝑛−1≤⋯≤𝜆₁≤0_𝑛𝜆=⎡⎢⎢⎢⎢⎢⎢⎢⎣+−+⋱−_𝑛+⎤⎥⎥⎥⎥⎥⎥⎥⎦,≥𝜆_𝑛−1≥⋯≥𝜆₁≥0𝐆^𝑛𝜆=⎡⎢⎢⎢⎢⎢⎢⎢⎣−−−⋱−₁−⎤⎥⎥⎥⎥⎥⎥⎥⎦≤𝜆₂≤⋯≤𝜆_𝑛≤0.(3.3)

Case 2. 𝐏_𝑘=𝐂_𝑘=cos𝜃^𝑘sin𝜃_𝑘𝐆_𝑛−sin𝜃_𝑘cos𝜃_𝑘⟹=⎡⎢⎢⎢⎢⎢⎢⎢⎣+−+⋯(−1)^𝑛−1−+−⋯(−1)^𝑛−2(−⋯1)_𝑛−2⋯⋯⋯⋯(−1)_𝑛−1⋯⋯+−𝜆⋯⋯−+₁⎤⎥⎥⎥⎥⎥⎥⎥⎦,≥𝜆₂≥⋯≥𝜆_𝑛≥0𝐆^𝑛=⎡⎢⎢⎢⎢⎢⎢⎢⎣−−+⋯(−1)^𝑛−1−−−⋯(−1)^𝑛−2(−⋯1)_𝑛−2⋯⋯⋯⋯(−1)_𝑛−1⋯⋯−−𝜆⋯⋯−−_𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎦,𝐆≤𝜆_𝑛−1≤⋯≤𝜆₁≤0_𝑛=⎡⎢⎢⎢⎢⎢⎢⎢⎣++−⋯(−1)^𝑛+++⋯(−1)^𝑛−1(−⋯1)_𝑛−1⋯⋯⋯⋯(−1)_𝑛⋯⋯++𝜆⋯⋯++_𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎦,≥𝜆_𝑛−1≥⋯≥𝜆₁≥0𝐆^𝑛=⎡⎢⎢⎢⎢⎢⎢⎢⎣−+−⋯(−1)^𝑛+−+⋯(−1)^𝑛−1(−⋯1)_𝑛−1⋯⋯⋯⋯(−1)_𝑛⋯⋯−+𝜆⋯⋯+−₁⎤⎥⎥⎥⎥⎥⎥⎥⎦≤𝜆₂≤⋯≤𝜆_𝑛≤0.(2)

Case 3. 𝐏_𝑘=𝐃_𝑘=−cos𝜃^𝑘sin𝜃_𝑘𝐆_𝑛sin𝜃_𝑘cos𝜃_𝑘⟹=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝜆⋯+++−⋯+++−⋯+++−⋯⋯⋯⋯⋯⋯+++−⋯−−−+₁⎤⎥⎥⎥⎥⎥⎥⎥⎦,≥𝜆₂≥⋯≥𝜆_𝑛≥0𝐆^𝑛=𝜆⎡⎢⎢⎢⎢⎢⎢⎢⎣⋯−++−⋯+−+−⋯⋯⋯++⋯⋯⋯⋯⋯⋯++−−⋯−−−−_𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎦,𝐆≤𝜆_𝑛−1≤⋯≤𝜆₁≤0_𝑛=𝜆⎡⎢⎢⎢⎢⎢⎢⎢⎣⋯+−−+⋯−+−+⋯⋯⋯−−⋯⋯⋯⋯⋯⋯−−++⋯++++_𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎦,≥𝜆_𝑛−1≥⋯≥𝜆₁≥0𝐆^𝑛=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝜆⋯−−−+⋯−−−+⋯⋯⋯−−⋯⋯⋯⋯⋯⋯−−−+⋯++++₁⎤⎥⎥⎥⎥⎥⎥⎥⎦.≤𝜆₂≤⋯≤𝜆_𝑛≤0(3.5)

Proof. If 𝜃_𝑘∈[0,𝜋/2], then the signs of 𝑎_𝑘 and 𝑐_𝑘 depend solely on selection of canonic orthogonal matrices 𝐏_𝑘(𝑘=1,…,𝑛−1). For any sign of sequence {𝜆_𝑚}(𝑚=1,…,𝑛) and its monotonicy realized through enumeration of its members, one can readily check the sign patterns stated above: by using (2.18) to determine signs of the diagonal entries in 𝐆_𝑛 and by using Lemma 3.1 or Lemma 3.2 to determine signs of 𝜀_𝑘(𝑘=1,…,𝑛−1). Observe that only in Case 1 when 𝜆_𝑛≥𝜆_𝑛−1≥⋯≥𝜆₁≥0, that is, when the sequence {𝜆_𝑚}(𝑚=1,…,𝑛) is nonnegative and increasing (but not strictly), matrix 𝐆_𝑛 is produced with hd sign pattern, including the possible presence of zero entries. 𝐆_𝑛 may attain a sparse structure if, for example, some eigenvalues are equal. To see that, let us firstly suppose 𝜆₁=⋯=𝜆_𝑘=𝜆. Then from (2.17)-(2.18) it follows that 𝜆^∗1=⋯=𝜆^∗𝑘=𝜆,𝜓₁₁=⋯=𝜓_{𝑘−1,𝑘−1}=𝜆 and 𝜀₁=⋯=𝜀_𝑘−1=0, thus obviously making the matrix 𝐆_𝑛 (2.21) with sparse structure. By using (2.17)-(2.18), (2.21) and both Lemmas, we can readily infer that if 𝜃_𝑘∈(0,𝜋/2)(𝑘=1,…,𝑛−1) and the sequence {𝜆_𝑚}(𝑚=1,…,𝑛) is strictly monotone, then matrix 𝐆_𝑛 (2.21) is produced with no zero entries in all three considered cases.

Remark 3.4. Let 𝜆₁𝜆₂⋯𝜆_𝑛≠0. Then, since 𝐆_𝑛=𝐔𝐆₁𝐔^T and 𝐔⁻¹=𝐔^T (recall that U is orthogonal), it follows that 𝐆^𝑛−1=(𝐔^T)⁻¹𝐆¹⁻¹𝐔⁻¹=𝐔𝐆¹⁻¹𝐔^T. Also, when the sequence {𝜆_𝑚}(𝑚=1,…,𝑛) is increasing (decreasing), then the sequence {𝜆^𝑚−1}(𝑚=1,…,𝑛) is decreasing (increasing). These facts and Theorem 3.3 offer a possibility of determining the sign pattern of 𝐆^𝑛−1 without really inverting 𝐆_𝑛. Furthermore, by using (2.17)-(2.18), (2.21), 𝐆^𝑛−1 can be calculated explicitly, also without really inverting 𝐆_𝑛.

Theorem 3.5. Let the positive increasing sequence {𝜆_𝑚}(𝑚=1,…,𝑛) be the spectrum of 𝐆_𝑛 produced by using Case 1 of Theorem 3.3. Then there always exists such a diagonal matrix 𝐃∶=𝑑𝑖𝑎𝑔(𝑑₁,𝑑₂,…,𝑑_𝑛) with positive diagonal entries which makes 𝐃𝐆_𝑛𝐃 truly hyperdominant.

Proof. If 𝐆₁∶=diag(𝜆₁,𝜆₂,…,𝜆_𝑛), then by Case 1 of Theorem 3.3, the nonsingular matrix 𝐆_𝑛=𝐔𝐆₁𝐔^T will have hd sign pattern and by Remark 3.4𝐆^𝑛−1=𝐔𝐆¹⁻¹𝐔^T will be nonnegative matrix. Since 𝑑_𝑚>0(𝑚=1,…,𝑛), then the nonsingular symmetric matrix 𝐃𝐆_𝑛𝐃 is produced with hd sign pattern, but it may not be truly hd, unless hd margin of each of its rows (or columns) is nonnegative (recall that hd margin of a row or a column is sum of all entries in that row or column). If 𝐆_𝑛=[𝑔_𝑘𝑚](𝑘,𝑚=1,…,𝑛), then hd margin 𝑝_𝑘 of the 𝑘th row (or the 𝑘th column) in 𝐃𝐆_𝑛𝐃 is given by 𝑝_𝑘=^𝑛_𝑚₌₁𝑔_𝑘𝑚𝑑_𝑚𝑑_𝑘=𝑑_𝑘𝛼_𝑘,where𝛼_𝑘∶=^𝑛_𝑚₌₁𝑔_𝑘𝑚𝑑_𝑚,𝑘=1,…,𝑛.(3.6) Let we arbitrarily select 𝛼_𝑘>0(𝑘=1,…,𝑛) and let 𝐚∶=[𝛼₁𝛼₂⋯𝛼_𝑛T𝛼¹𝛼₂⋯𝛼_𝑛^T,col(𝐃)∶=[𝑑₁𝑑₂⋯𝑑_𝑛]T𝑑₁𝑑₂⋯𝑑_𝑛]^T and 𝐩∶=[𝑝₁𝑝₂⋯𝑝_𝑛]T𝑝₁𝑝₂⋯𝑝_𝑛]^T. Then, from (3.6) it follows that 𝐆_𝑛col(𝐃)=𝐚, that is, col(𝐃)=𝐆^𝑛−1𝐚>𝟎_𝑛,1 and 𝐩=𝐃𝐚>𝟎_𝑛,1. This not only means that 𝐃𝐆_𝑛𝐃 has hd sign pattern, but that it is truly hd furthermore. Obviously, as much as “𝛼”s are assumed greater, the greater will be row (column) hd margins of 𝐃𝐆_𝑛𝐃. This completes the proof of theorem.

4. Explicit Solution of IEP of Hd Matrices with Uncommitted and with Assigned Nonnegative Spectrum

Theorem 4.1. Let 𝜃_𝑘(𝑘=1,…,𝑛−1) be a set of angles selected from the range [0,𝜋/2] and let {𝜆₁,𝜆₂,…,𝜆_𝑛} be uncommitted nonnegative spectrum of the real symmetric matrix 𝐆_𝑛=𝐔𝐆₁𝐔^𝑇[𝐆₁=𝑑𝑖𝑎𝑔(𝜆₁,𝜆₂,…,𝜆_𝑛)] which is to be produced as truly hd. Suppose that through enumeration of eigenvalues the sequence {𝜆_𝑚}≥0(𝑚=1,…,𝑛) is made increasing. Then, matrix 𝐆_𝑛 given by (2.21) will be truly hyperdominant if 𝜆₁ is sufficiently great.

Proof. Since by assumption the conditions of Theorem 3.3 (Case 1) are satisfied, then 𝐆_𝑛 produced by using (2.21) has hd sign pattern. As it is 𝜀_𝑘=(𝜆^∗𝑘−𝜆_𝑘+1)𝑎_𝑘(𝑘=1,…,𝑛−1), then from (2.17)-(2.18), (2.21) it follows that hd margin 𝑝_𝑚 of the 𝑚th row (or column) from 𝐆_𝑛(𝑚=1,…,𝑛) can be in general represented as 𝑝_𝑚=𝛼^1(𝑚)𝜆₁𝑝+𝛼^2(𝑚)𝜆₂+⋯+𝛼^𝑚(𝑚)𝜆_𝑚+𝛼^(𝑚)𝑚+1𝜆_𝑚+1(𝑚=1,…,𝑛−1),_𝑛=𝛼^1(𝑛)𝜆₁+𝛼^2(𝑛)𝜆₂+⋯+𝛼^{(𝑛)𝑛−1}𝜆_𝑛−1+𝛼^𝑛(𝑛)𝜆_𝑛,(4.1) where “𝛼” coefficients are defined as follows: 𝛼𝛼¹⁽¹⁾∶=𝑎₁𝑎₁+𝑎₂𝑐₁+𝑎₃𝑐₂𝑐₁+⋯+𝑎_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑐₁+𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑐₁,𝑚=1,₂₍₁₎𝛼_1(𝑚)∶=𝑏₁𝑏₁+𝑎₂𝑑₁+𝑎₃𝑐₂𝑑₁+⋯+𝑎_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑑₁+𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑑₁,∶=𝑎_𝑚𝑐_𝑚−1⋯𝑐₂𝑐₁𝑎₁+𝑎₂𝑐₁+𝑎₃𝑐₂𝑐₁+⋯+𝑎_𝑚𝑐_𝑚−1⋯𝑐₂𝑐₁𝛼_2(𝑚)+⋯+𝑎_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑐₁+𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑐₁,∶=𝑎_𝑚𝑐_𝑚−1⋯𝑐₂𝑑₁)(𝑏₁+𝑎₂𝑑₁+𝑎₃𝑐₂𝑐₁+⋯+𝑎_𝑚𝑐_𝑚−1⋯𝑐₂𝑑₁+⋯+𝑎_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑑₁𝛼+𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑑₁,𝑝=3,…,𝑚,_𝑝(𝑚)∶=𝑎_𝑚𝑐_𝑚−1⋯𝑐_𝑝𝑑_𝑝−1𝑏_𝑝−1+𝑎_𝑝𝑑_𝑝−1+𝑎_𝑝+1𝑐_𝑝𝑑_𝑝−1+⋯+𝑎_𝑛−1𝑐_𝑛−2⋯𝑐_𝑝𝑑_𝑝−1𝛼+𝑐_𝑛−1𝑐_𝑛−2⋯𝑐_𝑝𝑑_𝑝−1,𝑚=2,…,𝑛−1,𝑝=3,…,𝑚_𝑚(𝑚)𝛼_(𝑚)𝑚+1∶=𝑎_𝑚𝑑_𝑚−1𝑏_𝑚−1+𝑎_𝑚𝑑_𝑚−1+𝑎_𝑚+1𝑐_𝑚𝑑_𝑚−1+⋯+𝑎_𝑛−1𝑐_𝑛−2⋯𝑐_𝑚𝑑_𝑚−1+𝑐_𝑛−1𝑐_𝑛−2⋯𝑐_𝑚𝑑_𝑚−1,𝛼∶=𝑏_𝑚𝑏_𝑚+𝑎_𝑚+1𝑑_𝑚+𝑎_𝑚+2𝑐_𝑚+1𝑑_𝑚+⋯+𝑎_𝑛−1𝑐_𝑛−2⋯𝑐_𝑚+1𝑑_𝑚+𝑐_𝑛−1𝑐𝑚_𝑛−2=⋯2𝑐,…_𝑚+1,𝑑𝑛_𝑚−,2,_{𝑛(𝑛−1)}𝛼_1(𝑛)∶=𝑏_𝑛−1𝑏_𝑛−1+𝑑_𝑛−1,𝛼_2(𝑛)∶=𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑐₁𝑎₁+𝑎₂𝑐₁+𝑎₃𝑐₂𝑐₁+⋯+𝑎_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑐₁+𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑐₁,𝛼_𝑝(𝑛)∶=𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑑₁𝑏₁+𝑎₂𝑑₁+𝑎₃𝑐₂𝑑₁+⋯+𝑎_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑑₁+𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑑₁∶=𝑐_𝑛−1𝑐_𝑛−2⋯𝑐_𝑝𝑑_𝑝−1𝑏_𝑝−1+𝑎_𝑝𝑑_𝑝−1+𝑎_𝑝+1𝑐_𝑝𝑑_𝑝−1+⋯+𝑎_𝑛−1𝑐_𝑛−2⋯𝑐_𝑝𝑑_𝑝−1𝛼+𝑐_𝑛−1𝑐_𝑛−2⋯𝑐_𝑝𝑑_𝑝−1,𝑝=3,…,𝑛−1._𝑛(𝑛)∶=𝑑_𝑛−1𝑏_𝑛−1+𝑑_𝑛−1.(4.2) According to Case 1 of Theorem 3.3, both 𝑎_𝑘 and 𝑐_𝑘 are nonnegative when 𝜃_𝑘∈[0,𝜋/2](𝑘=1,…,𝑛−1). Then, from (4.2) we see that 𝛼^1(𝑚)≥0(𝑚=1,…,𝑛), whereas other “𝛼”s may be nonpositive. Since “𝛼”s depend only on selection of “𝜃”s, then by presuming 𝜆₁=𝜆₂=⋯=𝜆_𝑛=𝜆≠0, we obtain from (2.21) 𝐆_𝑛=𝜆𝐈_𝑛 and 𝑝_𝑚=𝜆(𝑚=1,…,𝑛) and from (4.1) we conclude that in general it holds: ^𝑚⁺¹_𝑝=1𝛼^𝑝(𝑚)≡1(𝑚=1,…,𝑛−1),^𝑛_𝑞=1𝛼^𝑞(𝑛)≡1.(4.3) Although 𝐆_𝑛 is produced with hd sign pattern, it will not be truly hd unless each of its row (column) hd margins is nonnegative, that is, 𝐩=[𝑝₁𝑝₂⋯𝑝_𝑛]T𝑝₁𝑝₂⋯𝑝_𝑛]^T≥𝟎_𝑛,1. The column vector p with entries (4.1) can be written as 𝐩=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝛼¹⁽¹⁾𝛼𝛼²⁽¹⁾00⋯0₁₍₂₎𝛼⋯𝛼⋯²⁽²⁾𝛼⋯³⁽²⁾⋯0⋯⋯⋯_1(𝑛−1)0𝛼^2(𝑛−1)𝛼^3(𝑛−1)⋯⋯𝛼^{𝑛(𝑛−1)}𝛼^1(𝑛)𝛼^2(𝑛)𝛼^3(𝑛)⋯⋯𝛼^𝑛(𝑛)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−⋯10001−⋯10001⋯00100⋯⋯⋯⋯00⋯⋯⋯⋯⋯0⋯⋯⋯⋯⋯⋯−⋯00001−⋯00011⋯00001⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦⁻¹⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜆¹𝜆₂−𝜆₁𝜆₃−𝜆₂𝜆_𝑛−1⋯−𝜆_𝑛−2𝜆_𝑛−𝜆_𝑛−1=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1𝛼₂₍₁₎⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦100⋯0₃_𝑟=2⋯1⋯𝛼^𝑟(2)𝛼⋯³⁽²⁾⋯0⋯⋯⋯_𝑛0_𝑠=2𝛼^{𝑠𝑛(𝑛−1)}_𝑢=3𝛼^{𝑢(𝑛−1)}⋯⋯𝛼^{𝑛(𝑛−1)}1^𝑛_𝑡=2𝛼^{𝑡𝑛(𝑛)}_𝑤₌₃𝛼^𝑤(𝑛)⋯⋯𝛼^𝑛(𝑛)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜆₁𝜆₂−𝜆₁𝜆₃−𝜆₂𝜆_𝑛−1⋯−𝜆_𝑛−2𝜆_𝑛−𝜆_𝑛−1⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦.(4.4) From (4.4) we finally obtain 𝑝₁=𝜆₁+𝜆₂−𝜆₁𝛼²⁽¹⁾,𝑝₂=𝜆₁+𝜆₂−𝜆₁³_𝑟=2𝑝𝛼^𝑟(2)+𝜆₃−𝜆₂𝛼³⁽²⁾,…,_𝑛=𝜆₁+𝜆₂−𝜆₁^𝑛_𝑡=2𝛼^𝑡(𝑛)+𝜆₃−𝜆₂^𝑛_𝑤₌₃𝛼^𝑤(𝑛)+⋯+𝜆_𝑛−𝜆_𝑛−1𝛼^𝑛(𝑛).(4.5) Since “𝛼”s and “∑”s in (4.5) are not certainly nonnegative and since the sequence {𝜆_𝑚}(𝑚=1,…,𝑛) is increasing, then firstly by arbitrary selection of differences 𝜆₂−𝜆₁≥0,𝜆₃−𝜆₂≥0,…,𝜆_𝑛−1−𝜆_𝑛−2≥0 and 𝜆_𝑛−𝜆_𝑛−1≥0 and thereafter a sufficiently great 𝜆₁≥0, all hd margins 𝑝_𝑚(𝑚=1,…,𝑛) can be made nonnegative, that is, the matrix 𝐆_𝑛 can be always produced as truly hyperdominant. This completes the proof of this theorem.

Presentation of explicit solution to the IEP of truly hd matrices with assigned nonnegative spectrum is now in order. It has been proved in [16] that this IEP always has at least one solution and that infinitely many others can be produced thereof by using Givens rotations. Solution of that IEP is important in electrical network synthesis of driving-point immittance functions and matrices of both passive and active, common-ground, transformerless, two-element-kind 𝑅𝐿𝐶 networks and in generation of various classes of canonic and noncanonic equivalent realizations [19, 22]. In [16] we have proved the existence of solution to the IEP of hd matrices with assigned nonnegative spectrum, but here we shall present the explicit construction of solution matrix entries by using other arguments. This represents the explicit solution of the problem opened in [17].

Theorem 4.2. For any set of real nonnegative numbers {𝜆₁,𝜆₂,…,𝜆_𝑛} there always exists at least one (and infinitely many) 𝑛×𝑛 real symmetric hyperdominant matrices having these numbers as eigenvalues. In other words, IEP of symmetric hd matrices with assigned nonnegative spectrum always has at least one solution.

Proof. We will take the same assumptions as in Theorem 4.1, except for 𝜃_𝑘∈(0,𝜋/2)(𝑘=1,…,𝑛−1). Through enumeration of eigenvalues, the nonnegative sequence {𝜆_𝑚}(𝑚=1,…,𝑛) is made increasing. Then, according to Theorem 3.3 (Case 1), the symmetric matrix 𝐆_𝑛=𝐔𝐆₁𝐔^T[𝐆₁=diag(𝜆₁,𝜆₂,…,𝜆_𝑛)] with spectrum {𝜆₁,𝜆₂,…,𝜆_𝑛} and the entries determined by (2.21), is produced with hd sign pattern, no matter what selection of 𝜃_𝑘s(𝑘=1,…,𝑛−1) has been made. Observe that in Case 1𝑎_𝑘=cos𝜃_𝑘 and 𝑐_𝑘=sin𝜃_𝑘. To make 𝐆_𝑛=𝐔𝐆₁𝐔^T truly hd, we will prove the existence of such “𝜃”s that make all “𝛼”s (and hence all “𝑝”s) in (4.1) nonnegative. Let we introduce the following positive sequence {𝑀_𝑚}(𝑚=1,…,𝑛)𝑀₁∶=𝑎₁+𝑎₂𝑐₁+𝑎₃𝑐₂𝑐₁+⋯+𝑎_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑐₁+𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑐₁𝑀₂∶=𝑎₂+𝑎₃𝑐₂+𝑎₄𝑐₃𝑐₃+⋯+𝑎_𝑛−1𝑐_𝑛−2⋯𝑐₃𝑐₂+𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₃𝑐₂𝑀₃∶=𝑎₃+𝑎₄𝑐₃+𝑎₅𝑐₄𝑐₃+⋯+𝑎_𝑛−1𝑐_𝑛−2⋯𝑐₄𝑐₃+𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₄𝑐₃𝑀𝑀_{𝑛−2𝑛−1}∶=𝑎_𝑛−2+𝑎_𝑛−1𝑐_𝑛−2+𝑐_𝑛−1𝑐_𝑛−2,𝑀_𝑛∶=𝑎_𝑛−1+𝑐_𝑛−1,∶=1.(4.6) Then, by using (4.2) we obtain a consistent set of inequalities that ensure nonnegativity of all “𝛼”s in (4.1) 𝛼𝛼¹⁽¹⁾=𝑎₁𝑀₁≥0,𝛼²⁽¹⁾=1−𝑎₁𝑀₁=𝑐₁(𝑐₁−𝑎₁𝑀₂)≥0,(4.7)_1(𝑘)≥0𝛼,𝛼^𝑝(𝑘)≥0⟺−𝑐_𝑝−1+𝑎_𝑝−1𝑀_𝑝≥0(𝑝=2,…,𝑘),(4.8)_(𝑘)𝑘+1𝛼≥0⟺𝑐_𝑘−𝑎_𝑘𝑀_𝑘+1≥0(𝑘=2,…,𝑛−1),(4.9)_1(𝑛)≥0,𝛼^𝑞(𝑛)≥𝛼0⟺−𝑐_𝑞−1+𝑎_𝑞−1𝑀_𝑞≥0(𝑞=2,…,𝑛−1),(4.10)_𝑛(𝑛)𝛼≥0⟺−𝑐_𝑛−1+𝑎_𝑛−1≥0,(4.11)_{𝑛(𝑛−1)}=𝑏^2𝑛−1+𝑏_𝑛−1𝑑_𝑛−1=𝑐^2𝑛−1−𝑎_𝑛−1𝑐_𝑛−1=𝑐_𝑛−1𝑐_𝑛−1−𝑎_𝑛−1≥0⟺𝑐_𝑛−1−𝑎_𝑛−1≥0.(4.12) For 𝑝=2 from (4.8) we obtain 𝑀₂≥𝑐₁/𝑎₁ and from (4.7) 𝑀₂≤𝑐₁/𝑎₁. Then, 𝑀₂=𝑐₁/𝑎₁,𝛼²⁽¹⁾=0,𝛼¹⁽¹⁾=1−𝛼²⁽¹⁾=1,𝑀₁=1/𝑎₁ and 𝑝₁=𝜆₁. From (4.11)-(4.12) it follows that 𝑎_𝑛−1=𝑐_𝑛−1(⇔𝜃_𝑛−1=𝜋/4) and 𝛼^{𝑛(𝑛−1)}=𝛼^𝑛(𝑛)=0 [inequalty (4.12) is the same as (4.9) if 𝑘=𝑛−1(𝑀_𝑛=1)]. For 𝑘=2,…,(𝑛−2), we obtain from (4.9) 𝑀_𝑘+1≤𝑐_𝑘/𝑎_𝑘 and for 𝑞=3,…,(𝑛−1), we obtain from (4.10) 𝑀_𝑞≥𝑐_𝑞−1/𝑎_𝑞−1. To summarize, we have proved that: (a) 𝑀_𝑟=𝑐_𝑟−1/𝑎_𝑟−1, for 𝑟=2,…,(𝑛−1) and (b) {𝛼^1(𝑘)=1,𝛼^𝑠(𝑘)=0[𝑠=2,…,(𝑘+1)] and 𝑝_𝑘=𝜆₁}, for 𝑘=1,…,(𝑛−1). And finally, from (4.10) we obtain 𝛼^𝑞(𝑛)=0 for 𝑞=2,…,𝑛,𝛼^1(𝑛)=1 and 𝑝_𝑛=𝜆₁. Since the matrix 𝐆_𝑛 has hd sign pattern and each of its row (column) hd margins is equal to 𝜆₁≥0, then 𝐆_𝑛 is truly hd matrix. This completes the proof of the theorem.

Remark 4.3. It relates to calculation of entries of 𝐆_𝑛. In Theorem 4.2 it is proved that 𝑀₁=1/𝑎₁ and 𝑀_𝑘=𝑐_𝑘−1/𝑎_𝑘−1(𝑘=2,…,𝑛−1). It is assumed 𝑀_𝑛=1. Since 𝜃_𝑛−1=𝜋/4, then 𝑀_𝑛−1=𝑎_𝑛−1+𝑐_𝑛−1=cos𝜃_𝑛−1+sin𝜃_𝑛−1=√2. For 𝑤=1,…,𝑛−1 it follows from (4.6) 𝑀_𝑤=𝑎_𝑤+𝑐_𝑤𝑀_𝑤+1=𝑎_𝑤+𝑐_𝑤𝑐_𝑤𝑎_𝑤=1𝑎_𝑤=𝑎^2𝑤+𝑐^2𝑤𝑎^2𝑤=1+𝑐^𝑤𝑎_𝑤²𝑎=1+𝑀^2𝑤+1=2+𝑀^2𝑤+2=⋯=𝑛−𝑤−1+𝑀^2𝑛−1=√𝑛−𝑤+1,_𝑤=cos𝜃_𝑤=1√𝑛−𝑤+1,𝑐^𝑤=sin𝜃_𝑤=𝑛−𝑤𝑛−𝑤+1.(4.13) By using (2.17)-(2.18), (2.21), (4.13) we can easily calculate all entries of the (initial) hd matrix 𝐆_𝑛. Other hd matrices having the same spectrum can be produced thereof by application of Givens rotations, one at a time.

5. Application of the Obtained Results in Electrical Network Synthesis

It is well known that synthesis methods of passive, common-ground, transformerless, two-element-kind 𝑅𝐿𝐶 networks yield topological configurations which are severely restricted by the method chosen [19]. By using of the results above, a new class of non-canonic, driving-point immittance realizations of passive, common-ground, transformerless, two-element-kind 𝑅𝐿𝐶 networks with minimum number of both nodes and elements of one kind can be generated with possibility of reduction in number of elements of other kind. The network synthesis is always performed by using normalization of both the complex frequency 𝑠 and the impedance 𝑍(𝑠). If Ω is a selected normalization frequency, then the normalized frequency is 𝑠_𝑛=𝑠/Ω. Similarly, if 𝑅₀ is a selected normalization resistance, then the normalized impedance is 𝑍_𝑛(𝑠)=𝑍(𝑠)/𝑅₀. Thereby we achieve [20]: (a) lesser dispersion of coefficients in normalized functions and (b) dimensionless manipulation of quantities. The normalized resistance of resistor 𝑅 is 𝑅_𝑛∶=𝑅/𝑅₀. The normalized impedance of an inductor 𝐿 is 𝑍_𝐿𝑛(𝑠)=𝐿𝑠/𝑅₀=(𝐿Ω/𝑅₀)𝑠_𝑛=𝐿_𝑛𝑠_𝑛 (𝐿_𝑛∶=𝐿Ω/𝑅₀-normalized inductance). The normalized impedance of a capacitor 𝐶 is 𝑍_𝐶𝑛(𝑠)=1/(𝐶𝑅₀𝑠)=1/[(𝑅₀𝐶Ω)𝑠_𝑛]=1/(𝐶_𝑛𝑠_𝑛) (𝐶_𝑛∶=𝑅₀𝐶Ω-normalized capacitance). To physically realize a network after synthesis, a denormalization process must be performed. The actual parameter values of 𝑅𝐿𝐶 elements are calculated as follows: 𝑅=𝑅_𝑛𝑅₀,𝐿=𝐿_𝑛𝑅₀/Ω,𝐶=𝐶_𝑛/(𝑅₀Ω). From now on it will be assumed that normalized synthesis is being carried out, but the lower index “𝑛’’ we be dropped from component labels for brevity.

It is well known that if a real rational function in 𝑠 can be realized as 𝑅𝐿 driving-point impedance 𝑍_𝑅𝐿(𝑠), then it can be also realized as 𝑅𝐶 driving-point admittance 𝑌_𝑅𝐶(𝑠) [20]. And similarly, if it can be realized as 𝑅𝐿 driving-point admittance 𝑌_𝑅𝐿(𝑠), then it can also be realized as 𝑅𝐶 driving-point impedance 𝑍_𝑅𝐶(𝑠). The 𝐿𝐶∶𝑅𝐶 transformation turns the synthesis of 𝐿𝐶 driving-point impedance 𝑍^□𝐿𝐶(𝑠) to synthesis of 𝑅𝐶 driving-point impedance 𝑍_𝑅𝐶(𝑠)=𝑍^□𝐿𝐶(√𝑠)/√𝑠 [20]. It also turns the synthesis of 𝐿𝐶 driving-point admittance 𝑌^□𝐿𝐶(𝑠) to synthesis of 𝑅𝐶 driving-point admittance 𝑌_𝑅𝐶(𝑠)=√𝑠⋅𝑌^□𝐿𝐶(√𝑠). These 𝑅𝐶 driving-point imittances are at first realized by prototype 𝑅𝐶 networks and thereof are produced the desired 𝐿𝐶 networks in the following way: capacitors in 𝑅𝐶 and 𝐿𝐶 networks remain the same, but the resistor from 𝑅𝐶 network turns to inductor in 𝐿𝐶 network with the same parameter value. Also, 𝐿𝑅∶𝑅𝐶 transformation turns the synthesis of 𝑅𝐿 driving-point impedance 𝑍^□𝑅𝐿(𝑠) to synthesis of 𝑅𝐶 driving-point impedance 𝑍_𝑅𝐶(𝑠)=𝑍^□𝑅𝐿(𝑠)/𝑠. It also turns synthesis of 𝑅𝐿 driving-point admittance 𝑌^□𝑅𝐿(𝑠) to synthesis of 𝑅𝐶 driving-point admittance 𝑌_𝑅𝐶(𝑠)=𝑠𝑌^□𝑅𝐿(𝑠). These 𝑅𝐶 imittances are realized by prototype 𝑅𝐶 networks and the desired 𝑅𝐿 networks are produced thereof in the following way: the resistor from 𝑅𝐶 network turns to inductor in 𝐿𝑅 network with the same parameter value, and the capacitor from 𝑅𝐶 network turns to resistor in 𝐿𝑅 network with reciprocal parameter value. Bearing all the aforementioned on mind, we can obviously restrict our consideration only to synthesis of driving-point impedance functions 𝑍_𝑅𝐶(𝑠) of 𝑅𝐶 networks, which satisfy the following well known analytic necessary and sufficient conditions [20]: (a) 𝑍_𝑅𝐶(𝑠) is real rational function in 𝑠, (b) It has only simple poles on negative real axis, or at the origin. At infinity it cannot have pole and (c) Residues of these poles are real and positive and 𝐴_∞∶=lim_𝑠→∞𝑍_𝑅𝐶(𝑠)≥0.

In general, the first canonic Foster's expansion (form) of 𝑍_𝑅𝐶(𝑠) [20] reads𝑍_𝑅𝐶(𝑠)=𝐴_∞+^𝑛_𝑚₌₀𝐴_𝑚𝑠+𝑠_𝑚[𝐴_∞>0;𝑠₀=0,𝐴₀>0;𝑠_𝑝,𝐴_𝑝>0(𝑝=1,…,𝑛)],(5.1) where 𝐴_𝑚 is residue of the pole 𝑠_𝑚(𝑚=0,1,…,𝑛). The network which realizes driving-point impedance 𝑍_𝑅𝐶(𝑠) (5.1) with minimum number of nodes (=𝑛+1), minimum number of resistors (=𝑛+1) and minimum number of capacitors (=𝑛+1) is depicted in Figure 1. Observe that neither the resistors, nor the capacitors share common-node and hence the overall network realization is said to be non common-grounded.

Figure 1 

The first canonic Foster's realization of 𝑍𝑅𝐶(𝑠). Denoted are the normalized “values” of 𝑅𝐶 parameters.

Now, we will present our synthesis procedure. If for a given driving-point impedance 𝑍_𝑅𝐶(𝑠) we found that 𝐴₀>0 and/or 𝐴_∞>0, then in the preamble of the realization procedure 𝐴_∞ and/or 𝐴₀/𝑠 should be at first extracted from (5.1) and realized by a series connection of resistor 𝑅_∞=𝐴_∞ and capacitor 𝐶₀=1/𝐴₀, thereby leaving for realization the driving-point impedance 𝑍_𝑅𝐶(𝑠)=𝑍_𝑅𝐶(𝑠)−𝐴_∞−𝐴₀/𝑠 with solely 𝑛 poles lying on the negative real axis. In the sequel we will assume that 𝑍_𝑅𝐶(𝑠) has only 𝑛 such poles.

Let 𝐂=diag(𝐶₁,𝐶₂,…,𝐶_𝑛) and 𝐆=diag(𝐺₁,𝐺₂,…,𝐺_𝑛) be diagonal 𝑛×𝑛 matrices with strictly positive diagonal entries corresponding to the normalized capacitances and conductances, respectively. If we arbitrarily choose a nonsingular 𝑛×𝑛 matrix T, then the reciprocal passive networks which realize 𝐂𝑠+𝐆 and 𝐘(𝑠)=𝐓(𝐂𝑠+𝐆)𝐓^T will have the same natural frequencies. By arbitrary selection of 𝑛×𝑛 nonsingular diagonal matrices 𝜹=diag(𝛿₁,𝛿₂,…,𝛿_𝑛), a broad class of nonsingular 𝑛×𝑛 matrices 𝐓 can be generated with assumption 𝐓=𝐕𝜹𝐔, where 𝐔 and 𝐕 are 𝑛×𝑛 orthogonal matrices. Since 𝐘(𝑠)=𝐓(𝐂𝑠+𝐆)𝐓^T=𝐕[𝜹(𝐔𝐂𝐔^T)𝜹𝑠+𝜹(𝐔𝐆𝐔^T)𝜹]𝐕^T, then𝐙(𝑠)=[𝑧_𝑚𝑟(𝑠)]_𝑛×𝑛=𝐘⁻¹(𝑠)=[𝐓(𝐂𝑠+𝐆)𝐓^T]⁻¹=(𝐕𝜹⁻¹𝐔𝐂^1/2)(𝑠𝐈_𝑛+𝐆𝐂⁻¹)⁻¹(𝐕𝜹⁻¹𝐔𝐂^1/2)^T.(5.2)Various network topologies can be produced by different choices of 𝐔 and 𝐕. But, only by selecting 𝐂=𝐶𝐈_𝑛(𝐶>0) and 𝐕=𝐈_𝑛, the networks with minimum number of common-ground capacitors are produced; and only by selecting 𝐆=𝐺𝐈_𝑛(𝐺>0) and 𝐕=𝐈_𝑛, the networks with minimum number of common-ground resitors are produced. Let us select 𝐂=𝐶𝐈_𝑛(𝐶>0) and 𝐕=𝐈_𝑛, and let us assume in (5.1): 𝐴₀=𝐴_∞=0 and 𝑠_𝑛>𝑠_𝑛−1⋯>𝑠₁>0. Since 𝐔=[𝑢_𝑚𝑟](𝑚,𝑟=1,…,𝑛), then from (5.2) it follows that𝐘(𝑠)=𝐶𝜹²𝑠+𝜹(𝐔𝐆𝐔^T)𝜹,𝑧_𝑚𝑟(𝑠)=1𝐶𝛿_𝑚𝛿_{𝑟𝑛𝑝=1}𝑢_𝑚𝑝𝑢_𝑟𝑝𝑠+𝐺_𝑝/𝐶(𝑚,𝑟=1,…,𝑛).(5.3) The matrices which are effectively realized by common-ground network with 𝑛+1 nodes [(𝑛+1)th node is the common-ground] are 𝐶𝜹²𝑠 and 𝜹(𝐔𝐆𝐔^T)𝜹, provided that both are truly hd. According to (5.1) and (5.3) it holds 𝐺_𝑝=𝐶𝑠_𝑝(𝑝=1,…,𝑛) and 𝐺_𝑛>𝐺_𝑛−1⋯>𝐺₁>0. By using Theorem 3.3 [Case 1, 𝐏_𝑘=𝐀_𝑘 and 𝜃_𝑘∈(0,𝜋/2)(𝑘=1,…,𝑛−1)] we infer that 𝐔𝐆𝐔^T (2.21) is produced with hd sign pattern and no zero entries and with strictly positive inverse. Matrix 𝐔 (2.26) is lower Hessenberg with nonnegative entries, except for negative “𝑏”s. The same conclusions relating to 𝐔𝐆𝐔^T and 𝐔 also hold if we apply Case 1 of Theorem 3.3 with 𝐏_𝑘=𝐁_𝑘 and 𝜃_𝑘∈(0,𝜋/2)(𝑘=1,…,𝑛−1), except for “𝑑”s in (2.26) are then negative and “𝑏”s are positive. To realize 𝑍_𝑅𝐶(𝑠) we must select in (5.3) either 𝑚=𝑟=𝑛−1 or 𝑚=𝑟=𝑛, thus obtaining either 𝑍_𝑅𝐶(𝑠)=𝑧_{𝑛−1,𝑛−1}(𝑠), or 𝑍_𝑅𝐶(𝑠)=𝑧_𝑛𝑛(𝑠). By assuming 𝑚=𝑟=𝑛[⇔𝑍_𝑅𝐶(𝑠)=𝑧_𝑛𝑛(𝑠)],𝐏_𝑘=𝐀_𝑘 and 𝜃_𝑘∈(0,𝜋/2)(𝑘=1,…,𝑛−1), it follows from (2.26), (5.1), (5.3)𝐶𝛿^2𝑛=1∑^𝑛𝑝=1𝐴_𝑝,𝑎_𝑖−1=𝑑_𝑖−1=𝐴^𝑖∑^𝑖𝑞=1𝐴_𝑞^1/2,𝑐_𝑖−1=∑^{𝑖−1𝑟=1}𝐴_𝑟∑^𝑖𝑞=1𝐴_𝑞^1/2,(𝑖=2,…,𝑛).(5.4) To prove the existence of a physical realization of both 𝐶𝜹²𝑠 and 𝜹(𝐔𝐆𝐔^T)𝜹 we still have to determine the positive column vector col(𝜹)=[𝛿₁𝛿₂⋯𝛿_𝑛]T𝛿₁𝛿₂⋯𝛿_𝑛]^T which, according to Theorem 3.5, makes 𝜹(𝐔𝐆𝐔^T)𝜹 truly hd with possibly zero hd margins of at most 𝑛−1 rows. Let hd margin of the 𝑖th row in 𝜹(𝐔𝐆𝐔^T)𝜹 be 𝑝_𝑖(𝑖=1,…,𝑛) and let 𝐩∶=[𝑝₁𝑝₂⋯𝑝_𝑛]T𝑝₁𝑝₂⋯𝑝_𝑛]^T. If 𝐮∶=[11⋯1]T11⋯1]^T (𝑛 unities), then 𝐩=𝜹(𝐔𝐆𝐔^T)𝜹𝐮. Let we introduce a column vector 𝐩^∗∶=[𝑝^∗1𝑝^∗2⋯𝑝^∗𝑛]T𝑝^∗1𝑝^∗2⋯𝑝^∗𝑛]^T (‖𝐩^∗‖_E>0) of arbitrarily assumed real nonegative numbers. Since (𝐔𝐆𝐔^T)⁻¹ is strictly positive, then it always can be find a diagonal matrix 𝜹=diag(𝛿₁,𝛿₂,…,𝛿_𝑛) with positive diagonal entries, such that 𝐩^∗=(𝐔𝐆𝐔^T)𝜹𝐮=(𝐔𝐆𝐔^T)col(𝜹). Herefrom, we obtain col(𝜹)=(𝐔𝐆𝐔^T)⁻¹𝐩^∗, that is, that 𝛿_𝑖>0(𝑖=1,…,𝑛). Since 𝐩=𝜹(𝐔𝐆𝐔^T)𝜹𝐮=𝜹𝐩^∗, then it follows 𝑝_𝑖=𝛿_𝑖𝑝^∗𝑖≥0(𝑖=1,…,𝑛), bearing on mind that at most 𝑛−1 “𝑝”s can be equal to zero. These “𝑝”s indices correspond to indices of those rows (or columns) in 𝜹(𝐔𝐆𝐔^T)𝜹 which have zero hd margins. Then, from the overall network vanish resistors connecting common-ground to nodes with the same indices as that of rows (columns) with zero hd margins [22]. For different selections of 𝐩^∗, different algorithms and different topologically and parametrically equivalent realizations emerge. For example, if we select col(𝜹)=𝜇[𝑢₁₁𝑢₂₁⋯𝑢_𝑛1]T𝑢₁₁𝑢₂₁⋯𝑢_𝑛1]^T(𝜇>0), where it is according to (2.26), (5.4)𝑢₁₁=𝐴²𝐴₁+𝐴₂^1/2,𝑢_𝑝1=𝐴¹𝐴_𝑝+1(∑^𝑛𝑞=1𝐴_𝑞)(∑^𝑝+1𝑟=1𝐴_𝑟)^1/2(𝑝=2,…,𝑛−1),𝑢_𝑛1=𝐴¹∑^𝑛𝑞=1𝐴_𝑞^1/2,(5.5) then the column vector 𝐩 of row (column) hd margins of matrix 𝜹(𝐔𝐆𝐔^T)𝜹 with hd sign pattern reads𝐩=𝜹(𝐔𝐆𝐔^T)𝜹𝐮=𝜹(𝐔𝐆𝐔^T)col(𝜹)=𝐺₁col(𝜹²)>𝟎_𝑛,1.(5.6) This means that 𝜹(𝐔𝐆𝐔^T)𝜹 is truly hd. We will now present two algorithms for realization of driving-point impedances 𝑍_𝑅𝐶(𝑠) which rely on the results developed above.

Algorithm 1. Realization of 𝑍_𝑅𝐶(𝑠) with minimum number of common-ground capacitors and non-reduced number of resistors

(1⁰) Commencing with 𝐴_𝑖(𝑖=1,…,𝑛) calculate the entries of 𝐔, by using (2.26) and (5.4).

(2⁰) Arbitrarily select some 𝜇>0 and then calculate 𝐶=1/𝜇²𝐴₁ and 𝐺_𝑞=𝐶𝑠_𝑞(𝑞=1,…,𝑛).

(3⁰) Calculate col(𝜹)=𝜇[𝑢₁₁𝑢₂₁⋯𝑢_𝑛1]T𝑢₁₁𝑢₂₁⋯𝑢_𝑛1]^T, by using (5.5) and the entries of hd matrix 𝜹(𝐔𝐆𝐔^T)𝜹. Calculate 𝐩, by using (5.6).

(4⁰) Realize 𝜹(𝐔𝐆𝐔^T)𝜹 by common-ground, transformerless, conductance network. This can be done easily, almost by visual inspection of 𝜹(𝐔𝐆𝐔^T)𝜹 [22]. Attach to the ports of that network, enumerated by 1, 𝑝(𝑝=2,…,𝑛−1) and 𝑛, the common-ground capacitors with normalized capacitances𝐶₁=𝐴²𝐴₁(𝐴₁+𝐴₂),𝐶^𝑝=𝐴^𝑝+1(∑^𝑛𝑞=1𝐴_𝑞)(∑^𝑝+1𝑟=1𝐴_𝑟)(𝑝=2,…,𝑛−1),𝐶^𝑛=1∑^𝑛𝑞=1𝐴_𝑞,(5.7) respectively. The 𝑛th port of the overall network realizes driving-point impedance 𝑍_𝑅𝐶(𝑠), provided that all other ports are left open-circuited.

Algorithm 2. Realization of 𝑍_𝑅𝐶(𝑠) with minimum number of common-ground capacitors and the reduced number of resistors
(1⁰) The same as step (1⁰) of Algorithm 1. Let 𝐒=diag(𝑠₁,𝑠₂,…,𝑠_𝑛). Recall that 𝑠_𝑛>𝑠_𝑛−1⋯>𝑠₁>0.
(2⁰) Select 𝜀₁>0 and 𝜀_𝑖=0(𝑖=2,…,𝑛−1). Thereafter, by using (2.21) and (5.4), calculate 𝐶=[𝜀₁(𝑠¹⁻¹−𝑠²⁻¹)𝑐_𝑛−1𝑐_𝑛−2⋯𝑐₂𝑐₁𝑎₁]^2𝑛_𝑞=1𝐴_𝑞,𝐺_𝑖=𝐶𝑠_𝑖(𝑖=1,…,𝑛).(5.8)Calculate col(𝜹)=𝜀₁𝐶⁻¹(𝐔𝐒⁻¹𝐔^T)𝐞₁, where 𝐞₁=[10⋯0]T10⋯0]^T is 𝑛-dimensional column vector.
(3⁰) Calculate the entries of 𝜹(𝐔𝐆𝐔^T)𝜹 and its hd margin 𝑝₁=𝛿₁𝜀₁, where 𝛿₁=𝜀₁𝐶⁻¹𝐞^T₁𝐓𝐒⁻¹𝐓^T𝐞₁. Set for other hd margins 𝑝_𝑖=0(𝑖=2,…,𝑛).
(4⁰) Realize 𝜹(𝐔𝐆𝐔^T)𝜹 by common-ground, transformerless, conductance network and attach to its 𝑖th port the common-ground capacitor with normalized capacitance 𝐶𝛿^2𝑖(𝑖=1,…,𝑛). The 𝑛th port of the overall network realizes driving-point impedance 𝑍_𝑅𝐶(𝑠), provided that all other ports are left open-circuited.

5.1. A Numerical Example

Consider realization of the real rational function 𝑍^□𝐿𝐶(𝑠) as driving-point impedance of common-ground transformerless 𝐿𝐶 network with minimum number of capacitors and the reduced number of inductors𝑍^□𝐿𝐶(𝑠)=(𝑠²+1)(𝑠²+3)(𝑠²+5)(𝑠²+8)𝑠(𝑠²+2)(𝑠²+4)(𝑠²+6).(5.9)This function satisfies the necessary and sufficient conditions for driving-point immittances of 𝐿𝐶 networks: (a) it is an odd real rational function in 𝑠; (b) it has only simple poles located strictly on imaginary axis; and (c) residues of those poles are real and positive. Therefore, 𝑍^□𝐿𝐶(𝑠) can be realized both in two Foster's and in two Cauer's canonic forms [20]. The partial fraction expansion of 𝑍^□𝐿𝐶(𝑠) reads𝑍^□𝐿𝐶(𝑠)=(𝑠²+1)(𝑠²+3)(𝑠²+5)(𝑠²+8)𝑠(𝑠²+2)(𝑠²+4)(𝑠²+6)=𝑠+25𝑠+𝑠9²𝑠+/82+𝑠3²𝑠+/44+𝑠5²𝑠+/86.(5.10) The reactance function corresponding to 𝑍^□𝐿𝐶(𝑠) is 𝑋^□𝐿𝐶(𝜔)∶=𝑍^□𝐿𝐶(𝑗𝜔)/𝑗 and it is depicted in Figure 2. The first canonic Foster's realization of 𝑍^□𝐿𝐶(𝑠) with minimum number of nodes, noncommon-ground capacitors and inductors is depicted in Figure 3. Thereon are denoted the normalized values of 𝐿𝐶 parameters.

Figure 2 

The reactance function of 𝐿𝐶 network from the example.

Figure 3 

Driving-point impedance 𝑍□𝐿𝐶(𝑠) from example synthesized by using the first canonic Foster's form.

In Figure 4, it is depicted the first canonic Foster's realization of driving-point impedance 𝑍^□𝐿𝐶(𝑠) from Figure 3 with selected normalization frequency Ω=10⁶[rad/s] and selected normalization resistance 𝑅₀=10³[kΩ]. The network is excited by a sinusoidal current generator having constant current amplitude and discretely varying frequency 𝑓=𝜔/(2𝜋) within the range 𝑓∈[0.1,500] kHz. If the complex representative of generator current is 𝐼_𝑔 and the complex representative of the voltage across its terminals is 𝑈_𝑔, then the complex driving-point impedance of the overall 𝐿𝐶 network is 𝑍^□𝐿𝐶=𝑈_𝑔/𝐼_𝑔. The modulus of 𝑍^□𝐿𝐶, that is, |𝑍^□𝐿𝐶| (usually called 𝐿𝐶 impedance) obtained through PSPICE simulation within the range 𝑓∈[0.1,500] kHz is depicted in Figure 5.

Figure 4 

Denormalized realization of driving-point impedance 𝑍□𝐿𝐶(𝑠) from Figure 3.

Figure 5 

Driving-point impedance |𝑍□𝐿𝐶| of 𝐿𝐶 network from Figure 4.

Now, we will realize 𝑍^□𝐿𝐶(𝑠) by using the proposed Algorithm 2. After 𝐿𝐶∶𝑅𝐶 transformation, we firstly produce the function 𝑍_𝑅𝐶(𝑠)=𝑍^□𝐿𝐶(√𝑠)/√𝑠=1+5/2𝑠+𝑍_𝑅𝐶(𝑠), where 𝑍_𝑅𝐶(𝑠) is driving-point impedance of 𝑅𝐶 network which should be expanded into partial fractions as follows:𝑍_𝑅𝐶(𝑠)=𝐴¹𝑠+𝑠₁+𝐴²𝑠+𝑠₂+𝐴³𝑠+𝑠₃||||||𝐴₁=98,𝐴²=34,𝐴³=5𝑠₁8,𝑠=2,𝑠₂=4,𝑠₃=6,₃>𝑠₂>𝑠₁>0,𝐒∶=diag(𝑠₁,𝑠₂,𝑠₃).(6.1) In step (1⁰) of Algorithm 2 we determine the orthogonal matrix 𝐔 by using 𝐴₁,𝐴₂, and 𝐴₃ (see (2.26) and (5.4))𝐔=⎡⎢⎢⎢⎢⎢⎢⎢⎣25−350320110−√3292031021⎤⎥⎥⎥⎥⎥⎥⎥⎦.(6.2)By assuming 𝜀₁=8√5/3 in step (2⁰), we further easily obtain 𝐶=1,𝐺₁=2,𝐺₂=4,𝐺₃=6 and col(𝜹)=[𝛿₁𝛿₂𝛿₃]T𝛿₁𝛿₂𝛿₃]^T=[2.0870.3650.632]T2.0870.3650.632]^T. In step (3⁰) we firstly calculate 𝜹(𝐔𝐆𝐔^T)𝜹 and then hd margins of its rows (columns),𝜹(𝐔𝐆𝐔^T)𝜹=⎡⎢⎢⎢⎣−−1301...931371739−−000.6..33972330−−101.4..13312990⎤⎥⎥⎥⎦,𝑝₁=12.444,𝑝₂=𝑝₃=0.(6.3) In step (4⁰) we calculate the normalized capacitances of common-ground capacitors: 𝐶₁=4.355,𝐶₂=0.133 and 𝐶₃=0.400. Realization of driving-point impedance 𝑍_𝑅𝐶(𝑠) by transformerless, common-ground 𝑅𝐶 network with minimum number of nodes (=𝑛+1), reduced number of inductors and minimum number of common-ground capacitors (=𝑛), begins by realization of conductance matrix 𝜹(𝐔𝐆𝐔^T)𝜹 which can be accomplished almost by inspection of that matrix [22]. Then, to the 𝑖th port of the realized conductance network, it should be connected to the capacitor 𝐶_𝑖=𝛿^2𝑖(𝑖=1,2,3). The third port of the overall network realizes the 𝑅𝐶 driving-point impedance 𝑍_𝑅𝐶(𝑠), provided that all other ports are left open-circuited. By embedding in the third port a series connection of resistor and capacitor with the normalized parameter values 1 and 2/5, respectively, and by applying 𝑅𝐶∶𝐿𝐶 transformation thereafter, we finally produce noncanonic network which realizes 𝑍^□𝐿𝐶(𝑠) with minimum number of nodes and capacitors and with reduced number of inductors. That network is depicted in Figure 6 whereon are denoted the normalized (dimensionless) values of 𝐿𝐶 parameters.

Figure 6 

Driving-point impedance 𝑍□𝐿𝐶(𝑠) from the example synthesized by a noncanonic 𝐿𝐶 network.

In Figure 7, is depicted the noncanonic realization of driving-point impedance 𝑍^□𝐿𝐶(𝑠) from Figure 6 with selected normalization frequency Ω=10⁶[rad/s] and selected normalization resistance 𝑅₀=10³[kΩ]. The network is excited by a sinusoidal current generator with constant current amplitude and with discretely variable frequency 𝑓=𝜔/(2𝜋) within the range 𝑓∈[0.1,500] kHz. If the complex representative of generator current is 𝐼_𝑔 and the complex representative of the voltage across its terminals is 𝑈_𝑔, then the complex driving-point impedance of the overall 𝐿𝐶 network is 𝑍^□𝐿𝐶=𝑈_𝑔/𝐼_𝑔. The modulus of 𝑍^□𝐿𝐶, that is, |𝑍^□𝐿𝐶| (usually called 𝐿𝐶 impedance) obtained through PSPICE simulation within the range 𝑓∈[0.1,500] kHz is depicted in Figure 8. Since 𝐿𝐶 networks in Figures 4 and 7 are intentionally designed to be equivalent, then their driving-point impedances |𝑍^□𝐿𝐶| must have the same variations in frequency, as can be verified from Figures 5 and 8 qualitatively and more precisely by using numerical results of simulation.

Figure 7 

Denormalized realization of driving-point impedance 𝑍□𝐿𝐶(𝑠) from Figure 6.

Figure 8 

Driving-point impedance |𝑍□𝐿𝐶| of 𝐿𝐶 network from Figure 7.

6. Conclusions

A novel procedure for explicit construction of entries of real symmetric matrices with assigned spectrum is developed by using a group of four types of canonic, second-order, orthogonal transformations. It has been also shown that the orthogonal modal matrices corresponding to the produced real symmetrix matrices, are either lower or upper Hessenberg with explicitly constructed entries too. Thereafter, the inverse eigenvalue problems of real symmetric matrices with twelve specific types of sign patterns (including hyperdominant one) are explicitly solved providing that the signs of eigenvalues are the same (zeros are permitted) and that they are enumerated such as to establish the increasing or decreasing sequence. It is proved to arise thereof a possibility of explicit solving the inverse eigenvalue problem of symmetric hyperdominant matrices having either uncommitted or assigned nonnegative spectrum. The results obtained are then applied in synthesis of driving-point immittance functions of transformerless, common-ground, two-element-kind 𝑅𝐿𝐶 networks and in generation of their equivalent realizations with minimum number of nodes. The synthesis procedures proposed herein turn the synthesis problem of any immittance function of the two-element-kind 𝑅𝐿𝐶 network to the synthesis problem of impedance function of a prototype 𝑅𝐶 network.