A Note on the Square Roots of a Class of Circulant Matrices
We prove that any -circulant matrix and any even order skew -circulant matrix are diagonalizable for any . Then, we propose two algorithms for computing the square roots of the -circulant matrix and the skew -circulant matrix, respectively. In particular, we show that the square roots of the -circulant matrix are still -circulant matrices. Both the theoretical analysis and the numerical experiments show that our algorithms are faster than the standard Schur method.
Given a matrix , a matrix is called a square root of if . Matrix square roots appear in a variety of branches of mathematics, such as Markov models of finance, the solution of differential equations, the computation of the polar decomposition, and the matrix sign function . A number of methods have been proposed for computing the square roots of a matrix [2–9]. Among them, the Schur method  is the most popular and becomes the standard method for computing the matrix square roots. However, Schur method is not so efficient when the matrix order is relatively high. Thus, it is very desirable to design fast computing methods which can make full use of the particular properties of the matrices when the matrices possess special structures.
Circulant matrices and their generalizations have a wide range of applications in signal processing, coding theory, digital image disposal, self-regress design, Toeplitz systems, and so on [10–14]. A relatively comprehensive survey about circulant matrices can be found in . Recently, Lu and Gu  presented two efficient algorithms to compute the square roots of circulant matrices and quasi-skew circulant matrices, respectively. As they are based on LL iteration  and the modified Schulz iterative method, the two algorithms are faster than the standard Schur method. Subsequently, Mei generalized those methods and presented algorithms to compute the square roots of -circulant matrices and skew -circulant matrices . These algorithms are also faster than the standard Schur algorithm, but the work is restricted to the case in which the matrix is of even order and and can not be directly extended to compute the th root.
In this paper, we first show that -circulant matrices of any order for any complex number are diagonalizable and develop an algorithm to compute their principal square roots. Then, we show that skew -circulant matrices of even order for any complex number are diagonalizable, and develop an algorithm to compute their principle square roots. Both of our algorithms are theoretically and experimentally proved to be faster than the standard Schur method. Compared with the work in , our methods are more general in that they are valid for any -circulant matrix and any even order skew -circulant matrix, where can be any complex number. The remainder of this paper is organized as follows. In Section 2, we compute the square roots of -circulant matrices. In Section 3, we compute the square roots of skew -circulant matrices. In Section 4, we present two numerical experiments to exhibit the efficiency of the proposed algorithms in terms of the CPU time.
2. Square Roots of -Circulant Matrices
An complex matrix is called a -circulant matrix, where and . In particular, 1-circulant matrices are circulant matrices, and -circulant matrices are skew circulant matrices .
Another equivalent definition of a -circulant matrix is as follows : let be the set of all complex matrices, and then, is a -circulant matrix if and only if , where . In this section, we show that -circulant matrices are diagonalizable.
Lemma 1 (see ). If and are two -circulant matrices of the same order, then is also a -circulant matrix.
Lemma 2 (see ). If is a -circulant matrix, then for any , is also a -circulant matrix.
Lemma 3 (see ). Let ; then, is also a -circulant matrix. In particular, , where is the identity matrix.
Lemma 4 (see ). Let and let (, ); then, the eigenvalues of are where is the imaginary unit.
By (5), we can easily obtain the following result.
Corollary 8. The square roots of -circulant matrix are as follows:
Remark 9. We mention that the diagonalization methods [19–22] were used to compute the square roots of -circulant matrices, where is restricted to or . However, our diagonalization method is valid for any .
Next, we show that the square roots of a -circulant matrix are still -circulant matrices.
Theorem 10. Let (, ), let (), and let be defined as (6). For any diagonal matrix , is a -circulant matrix , where .
Proof. By Theorem 7, whether is a -circulant matrix depends on whether the following system of linear equations with unknown vector is consistent. Obviously the coefficient matrix of (8) is invertible. Thus, we have That is to say, is a -circulant matrix with the form of .
Algorithm 12. Compute a principal square root of a -circulant matrix .
Step 1. Compute the eigenvalues () of .
Step 2. Compute () such that (.
Step 3. Compute the inverse of .
Step 4. Compute .
Step 5. Compute the square root of .
Then, we obtain .
The cost of Step 1 is about flops by discrete Fourier transform . The cost of Step 2 is flops. The cost of Step 3 is about flops . The cost of Step 4 is about flops. The cost of Step 5 is about flops . So, it needs about flops in total. The algorithm has the same complexity as the diagonalization methods in [19–21]. But the methods therein are only concerned with the case that and . If we use the Schur method, it needs about flops in total . We also mention a related work in , which only needs about flops to compute the primary square root of a -circulant matrix. However, that work restricts the matrix to be of even order and to be of real number. Those restrictions are not needed in our algorithm.
3. Square Roots of Skew -Circulant Matrices
Let be an even number; then, is a skew -circulant matrix if , where (see ). Let (, ) and let
Lemma 13. Let . Then,(i);(ii) for odd number ; for even number .
Lemma 14. Let where (); then, .
Proof. The proof is obvious by paying attention to the fact that .
Lemma 15. A skew -circulant matrix of order can be written in the form of
Proof. Write and . Since we have ; that is, Then, Thus, we have that
Theorem 16. A skew -circulant matrix of order is diagonalizable.
Proof. Write ; then . By Lemma 13, Thus, we have By Theorem 7, there exists an invertible matrix defined in Lemma 14 such that where . Then, By (16), we have In order to obtain the eigenvalues and eigenvectors of , for , we assume that there exist , , and , such that Combining this with (20) gives Now, we use Lemma 14 to get Since and are linearly independent, we have So, , . Namely, Let ThenSo,
Corollary 17. The square roots of an even order skew -circulant matrix are as follows:
Proof. This is a direct result from (28).
Algorithm 18. Compute a principal square root of a skew -circulant matrix .
Step 1. Compute ().
Step 2. Compute the eigenvalues of : .
Step 3. Compute the square roots of whose arguments should be in .
Step 4. Compute (29).
Let . The cost of Step 1 is about flops by discrete Fourier transform . The cost of Step 2 is . The cost of Step 3 is . Since the matrix multiplication and inversion are equivalent in computational complexity , the cost of Step 4 is about . So, it needs about flops in total. If we use the Schur method  or the method in , it needs about flops in total. Moreover, our method allows to be a complex number, whereas the method in  only permits to be a real number.
4. Numerical Experiments
We present numerical experiments to compare the algorithms presented in this paper and the standard Schur method with respect to the execution time. The code was implemented in MATLAB 7.0 and run on a Windows-based machine with 2 GB of RAM and Intel Pentium Core Duo CPU running at 2.8 GHz.
We gradually increase the order of the matrix by 100 from 100 to 1100 and record the corresponding CPU time. Figure 1 shows the execution time for Algorithm 12 and the standard Schur method. Figure 2 compares the execution time for Algorithm 18 and the standard Schur method. From those results, we can confirm that our algorithms are clearly faster than the standard Schur method for computing the square roots of -circulant matrices and skew -circulant matrices.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is partly supported by the National Natural Science Foundation of China (61101228) and by the Fundamental Research Funds for the Central Universities (3132013337).
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