Maximum Principles for Normal Matrices

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Introduction and Main Results
Ky Fan maximum principle is a useful observation that characterizes an important property of hermitian matrices.It is interesting, therefore, to see whether it is possible to extend this rule to other types of matrices.In this note, we answer this question for normal matrices.The reader is referred to references [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] for detailed discussions of normal matrices and their properties.Let N = (n ij ) ∈ C n×n be a normal matrix with eigenvalues ν j , j = 1, . . ., n, that satisfy (1.1) Then N has a spectral decomposition of the form where S ∈ C n×n is a diagonal matrix and V ∈ C n×n is a unitary matrix whose columns are eigenvectors of N .The matrix V * denotes the conjugate transpose of V , and the term unitary matrix means that V * V = V V * = I.Let v j , j = 1, . . ., n, denote the jth column of V .Then (1.2) implies the equalities N v j = ν j v j for j = 1, . . ., n. (1.4) That is, v j is an eigenvector of N that corresponds to ν j .Let σ j , j = 1, . . ., n, denote the singular values of N arranged in decreasing order.Then the fact that N is a normal matrix implies the equalities Let 1 ≤ k ≤ n be a given positive integer.Then the matrices S k ∈ C k×k and V k ∈ C n×k are defined by the rules and In other words, S k is a principal submatrix of S, and V k is composed from the first k columns of V .Observe that (1.2) implies the equalities Note also that V k belongs to the set which contains all the n × k matrices that have orthonormal columns.
The maximum problems that we solve consider Rayleigh quotient matrices of the form Let N be a normal matrix as above, and let • be a unitarily invariant norm on C k×k .Then and the maximal value is obtained when Q = V k .
Recall that a matrix norm • on C n×n is called unitarily invariant if for any matrix A ∈ C n×n and any unitary matrix U ∈ C n×n we have the equalities A = U A = AU .The family of unitarily invariant norms includes several useful norms, such as Frobenius norm, the Schatten p-norms, Ky Fan k-norms, the trace norm, and the spectral norm.The relation between Theorem 1 and Ky Fan maximum principle can be seen by considering the trace norm • tr .Given a matrix A = (a ij ) ∈ C n×n its trace norm is defined as where σ j (A), j = 1, . . ., n, denote the singular values of A sorted in decreasing order.The conversion of Theorem 1 to handle the trace norm is simplified by applying the absolute-trace function The last result is the absolute trace theorem which was recently proved in [4].Assume for a moment that N is a positive semidefinite hermitian matrix.In this case ν j = σ j for j = 1, . . ., n, and the matrices Q * N Q are positive semidefinite.Hence the diagonal entries of these matrices are nonnegative and (1.13) is reduced to Ky Fan maximum principle [6] It is also easy to verify that the positive semidefinite requrement is not essential for Ky Fan maximum principle, so (1.14) holds whenever N is hermitian.However, if N is not positive semidefinite then the two problems may have different solutions.
Summarizing the above observations we see that both Theorem 1 and the absolute trace theorem can be viewed as extensions of Ky Fan maximum principle.The absolute trace theorem allows the use of normal matrices instead of hermitian matrices, while Theorem 1 achieves further extension by replacing the trace norm with any unitarily invariant norm.Below we will show that further extensions are gained by replacing the diagonal entries with eigenvalues and singular values. (1.15) Theorem 2. Let N ∈ C n×n be a normal matrix as above, and let • be a unitarily invariant norm on C k×k .Then and the maximal value is attained for V k .
As before, it is interesting to consider the trace norm.In this case (1.16) takes the form and, when N is a positive semidefinite hermitian matrix, (1.17) is reduced to The third maximum principle considers the singular values of Q * N Q.Let σ j (Q * N Q), j = 1, . . ., k, denote the singular values of Q * N Q arranged in decreasing order, and let σ(Q * N Q) denote the k × k diagonal matrix whose (j, j) diagonal entry equals (1.20) Then, here we consider the maximization of σ(Q * N Q .However, since the equality holds for any unitarily invariant norm, it is possible to replace Let N be a normal matrix as above, and let • be a unitarily invariant norm on C k×k .Then and the maximum value is attained for V k .
As in the former cases it is easy to verify that when using the trace norm on a positive semidefinite hermitian matrix the last assertion is reduced to Ky Fan maximum principle.It should be noted, however, that in the general case, when using an arbitrary unitarily invariant norm and N is an arbitrary normal matrix, the objective functions of the three maximum problems can be quite different.Yet, as we have seen, the three problems share the same solution matrix, V k , and the same optimal value, S k .
The rest of the paper continues as follows: The next section introduces the necessary theoretical basis, while Section 3 provides the proofs of Theorems 1-3.Finally, in Section 4 we expose interesting relations between the new maximum principles and a minimum norm problem that arises when searching a rank-k matrix that is closest to N .

Theoretical Background and Tools
We shall start by introducing some useful notations.Let A = (a ij ) ∈ C n×n be a given arbitrary matrix with eigenvalues λ j (A), j = 1, . . ., n, and singular values σ j (A), j = 1, . . ., n.Then there is no loss of generality in assuming that and These inequalities enable us to define the related diagonal matrices Thus, for example, λ(A) is an n × n diagonal matrix whose (j, j) entry equals λ j (A).Similarly, the matrix Next, we will say a few words about majorization and dominance.Let B ∈ C n×n and C ∈ C n×n be a pair of matrices with singular values (2.8) In this case we say that the singular values of B are majorized by those of C. The importance of this relation comes from Ky Fan dominance theorem [6], which says that (2.8) ensures the inequality for any unitarily invariant norm.
Another useful property stems from the interlacing theorems of Cauchy and Poincaré.The original statements of these theorems are about eigenvalues of hermitian matrices, e.g., [8] [11] [15].Yet when these theorems are adapted to singular values we have the following results.Let P k (A) denote the k × k principal submatrix of A which is obtained by deleting from A the last n − k columns and the last n − k rows.Let denote the singular values of P k (A).Then Moreover, let Q k be defined as in (1.9).Then the inequalities hold for all Q ∈ Q k .
Combining the interlacing relations with Ky Fan dominance theorem yields two powerful tools.First note that (2.12) means that the singular values of Q * AQ are majorized by those of S k (A), which yields the inequality The second tool is based on the inequality whose proof can be found, for example, in [3, p. 1237 The last inequalities mean that the singular values of δ(A) are majorized by those of A, which shows that δ(A) ≤ A . (2.17) Further results on majorization relations between δ(A), σ(A) and λ(A), can be found in [1], [4], [9, p. 176], [10, pp. 313-318] and [15, p. 262].

Proofs
In this section we provide the proofs of Theorems 1-3.
The proof of Theorem 1.Let Q be some matrix from Q k .Then (2.17) implies while from (2.13) we conclude that and Finally, from (1.8) we obtain that The proof of Theorem 2. Let Q be some matrix from Q k .Then Schur's triangularization theorem ensures the existence of a k × k unitary matrix, Q, such that where T ∈ C k×k is an upper-triangular matrix.Therefore, since the diagonal entries of T are eigenvalues of Q * N Q, Note also that the matrix belong to Q k , and Hence from (3.3) we conclude that while a further use of (1.8) shows that the above upper bound is achieved for V k .
The proof of Theorem 3. From (2.13) we obtain that is a rank-k truncated SVD of N that solves (4.1).
A second consequence of the spectral decomposition (1.1)-(1.8) is that the problem min is essentially equivalent to (4.1).This observation stems from the following facts.First note that for any Hence the optimal value of (4.3) exceeds that of (4.1).Yet for Q = V k problem (4.3) attains this value.This shows that both problems share the same optimal value, N − V k S k V * k , that V k solves (4.3), and that a solution for (4.When using this norm problem (4.1) is reduced to Eckart-Young lowrank approximation problem, e.g., [3, p. 1243] or [9, p. 217].Observe that for any Q ∈ Q k we have the equality which shows that the minimum norm problem min is equivalent to the maximum problem max The optimal values of (4.6) and (4. 3) provides a solution for (4.1).The relation between (4.3) and the maximum principle (1.22) is exposed by using the Frobenius matrix norm • F .Recall that for any matrix A = (a ij ) ∈ C n×n 7) are n j=k+1 |ν j | 2 and k j=1 |ν j | 2 , respectively, and the sum of these values equals N 2 F = n j=1 |ν j | 2 .DOI: 10.4236/alamt.2019.9300579 Advances in Linear Algebra & Matrix Theory )