Mathematical Problems in Engineering

Volume 2008, Article ID 513582, 25 pages

http://dx.doi.org/10.1155/2008/513582

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

^{1}Department of Physics & Electrical Engineering, Mechanical Engineering Faculty, University of Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia^{2}Electrical Engineering Faculty, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia

Received 20 January 2008; Accepted 22 May 2008

Academic Editor: Mohammad Younis

Copyright © 2008 D. B. Kandić and B. D. Reljin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 th-order hd matrix with assigned spectrum was the th-order orthogonal Hessenberg matrix constructed as a special product of 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 (, ) vanishes whenever .

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 networks and in generation of their equivalent realizations
[17–19]. networks are comprised
solely of resistors (), inductors (), and capacitors (). Driving-point immittance function of a lumped, time invariant,
linear electrical network is either a driving-point impedance ,
or a driving-point admittance ( is the complex frequency; , are real numbers; ). It is well known that a *real rational function* in can be driving-point immittance function
of network *if and only if* it is *positive
real* function in ; or similarly, a *necessary* condition for a stable square matrix of real rational
functions in to be driving-point immittance
matrix of a passive network is
that 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 networks and in generation of their equivalent realizations. The
two proposed realization procedures are illustrated by an example. Throughout
the paper denotes direct sum, denotes transpose of , bold capital letters denote matrices and stands for
the th-order unit or identity
matrix.

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

Let be assigned spectrum of the sought real symmetric matrices and let be spectral matrix. Consider a set of orthogonal matrices : which are either rotators ( and ) or reflectors ( and ). A useful set of orthogonal matrices is From the following two matrix recurrent relations we readily obtain real symmetric matrices and , which are both congruent and similar to Columns of the orthogonal modal matrix correspond to eigenvectors of . Out of different possibilities of using (2.3) in generation of and , only the two selected by (2.4) produce explicit expressions of entries of and in terms of and the entries of . 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 and we readily obtain and , by using (2.1) and (2.3): Let , , and , and let us firstly introduce in (2.5) the following notation: Thereafter, observing the partition of and obtained in (2.5) it can readily be anticipated the partition of subsequent matrices as follows: where is the symmetric matrix, is row vector, is matrix, is modified eigenvalue , and For from (2.1)–(2.3), (2.8) it follows that For , let us define: , , and thereafter and Then, from (2.8)-(2.9) it follows the identification which enables the partition of (2.10) in () to be like that of (2.10) in (), and that partition of (2.10) be rather simple Let . Having uncovered the partition pattern of , we can pursue partitioning of backwardly from to , by using (2.10)). Afterwards, we can produce , by using (2.10)-(2.11). The results are Since and then after defining and it follows from (2.12)-(2.13) Since and then on introducing , and we obtain from (2.14)-(2.15) the partition of which is amenable to the production of its entries in explicit form and is suitable for further discussion about solving some specific IEPs For , we consecutively obtain from and that generally it holds Since (2.17) and (2.17), then from () it follows that Observe that it is not necessary to calculate “”s from (2.18), but only the modified eigenvalues from (2.17) since it holds and . As it is , then for from (2.10)) it follows that The real symmetric matrix with assigned spectrum and the explicitly expressed entries can be derived from (2.16) and (2.20), bearing on mind that “”s and “”s are calculated by using , , modified eigenvalues (2.17) and : where denotes , denotes , denotes , and denotes . Entries of are , , and . They are calculated according to the following steps:

(a)Select arbitrarily the entries of orthogonal matrices , given by (2.1);(b)with , calculate the modified eigenvalues by using (2.17);(c)calculate and ;(d)calculate the entries of , by using (2.21).

Matrix (2.4) is orthogonal modal matrix established from eigenvectors of . We will now prove that is not only orthogonal, but also lower Hessenberg with explicitly expressed entries. Let us firstly produce and whose partition will enable us to anticipate the partition of If we now suppose that where is orthogonal upper Hessenberg matrix then since according to (2.2), it holds we may write further for By using (2.2), (2.23)-(2.24), it follows that and thereby it is proved our previous assumption that , where (2.23) is the orthogonal upper Hessenberg matrix with entries expressed explicitly. And finally, for from and (2.4), (2.23), we obtain and The entries of the orthogonal lower Hessenberg matrix are defined as follows: By using the similar arguments as in derivation of entries of matrix , the orthogonal matrix 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 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 be arbitrarily enumerated, thereby establishing the sequence . The nonnegative sequence will be denoted by , and the nonpositive one by Firstly, we will prove two lemmas.

Lemma 3.1. *If
the sequence is increasing [decreasing], then in (2.21) and the sequence .*

*Proof.*Since ,
then it is trivial to see from (2.17) and (2.18) that all diagonal entries of are nonnegative, that is, and no matter whether the sequence is increasing or decreasing. By virtue of orthogonality
of ,
we have .
If is *increasing* sequence, then for we have and for we obtain From (2.17) and
the last of inequalities (3.1) it follows and . If is *decreasing* sequence, then for we have and for we obtain From (2.17) and
the last of inequalities (3.2) it follows and . This
completes the proof of lemma. For a *nonpositive* sequence, an analogous lemma can be formulated.

Lemma 3.2. *If
the sequence
is
increasing [decreasing], then in (2.21) and the sequence .*

*Proof. *It is similar to that of Lemma 3.1, but
in this case the diagonal entries of are nonpositive, that is, and , no matter whether the sequence 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
are
arbitrarily selected angles from the range , then the entries of real symmetric
matrices with assigned spectrum , produced by (2.21), can attain the following twelve sign patterns (zero entries are permitted), depending
on selection of matrices (see (2.1)).*

*Case 1. *

*Case 2. *

*Case 3. *

*Proof. *If ,
then the signs of and depend solely on selection of canonic orthogonal
matrices .
For any sign of sequence 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 and by
using Lemma 3.1 or Lemma 3.2 to determine signs of .
Observe that only in Case 1 when that is, when the sequence is nonnegative and increasing (but not
strictly), matrix is produced with hd sign pattern, including the possible
presence of zero entries. may attain a sparse
structure if, for example, some eigenvalues are equal. To see that, let us firstly
suppose . Then from (2.17)-(2.18) it follows that and thus obviously making the matrix (2.21) with sparse structure. By using (2.17)-(2.18), (2.21) and both Lemmas, we can
readily infer that if and the
sequence is *strictly
monotone*, then matrix (2.21) is produced with no zero
entries in all three considered cases.

*Remark 3.4. *Let .
Then, since and (recall that **U** is orthogonal), it follows that .
Also, when the sequence is increasing (decreasing), then the sequence is decreasing (increasing). These facts and Theorem
3.3 offer a possibility of determining the sign pattern of without really inverting .
Furthermore, by using (2.17)-(2.18), (2.21), can be calculated explicitly, also without really
inverting .

Theorem 3.5. *Let
the positive increasing sequence be the
spectrum of produced by using Case 1 of Theorem 3.3. Then there always exists such
a diagonal matrix with positive
diagonal entries which makes truly hyperdominant.*

*Proof. *If , then by Case 1 of Theorem 3.3, the nonsingular matrix will have hd sign pattern and by Remark 3.4 will be nonnegative matrix. Since , then the nonsingular symmetric matrix 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 , then hd margin of the th row (or the th column) in is given by Let we arbitrarily
select and let and . Then, from (3.6) it follows that , that is, and . This not only
means that has hd
sign pattern, but that it is truly hd furthermore. Obviously, as much as “”s are assumed
greater, the greater will be row (column) hd margins of . 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 be a set of angles selected from the range and let be nonnegative spectrum of the real
symmetric matrix which is to
be produced as truly hd. Suppose that through enumeration of eigenvalues the sequence is made increasing. Then, matrix given by (2.21) will be truly hyperdominant if is
sufficiently great.*

*Proof. *Since by assumption the conditions
of Theorem 3.3 (Case 1) are satisfied, then produced by using (2.21) has hd
sign pattern. As it is ,
then from (2.17)-(2.18), (2.21) it follows that hd margin of the th
row (or column) from can be in
general represented as where “” coefficients
are defined as follows: According to Case 1 of Theorem 3.3, both and are nonnegative when . Then, from
(4.2) we see that ,
whereas other “”s may be nonpositive. Since “”s depend only
on selection of “”s, then by presuming , we obtain from (2.21) and and from (4.1) we conclude that in general it holds: