Perturbation Bounds for Eigenvalues and Determinants of Matrices. A Survey

: The paper is a survey of the recent results of the author on the perturbations of matrices. A part of the results presented in the paper is new. In particular, we suggest a bound for the difference of the determinants of two matrices which reﬁnes the well-known Bhatia inequality. We also derive new estimates for the spectral variation of a perturbed matrix with respect to a given one, as well as estimates for the Hausdorff and matching distances between the spectra of two matrices. These estimates are formulated in the terms of the entries of matrices and via so called departure from normality. In appropriate situations they improve the well-known results. We also suggest a bound for the angular sectors containing the spectra of matrices. In addition, we suggest a new bound for the similarity condition numbers of diagonalizable matrices. The paper also contains a generalization of the famous Kahan inequality on perturbations of Hermitian matrices by non-normal matrices. Finally, taking into account that any matrix having more than one eigenvalue is similar to a block-diagonal matrix, we obtain a bound for the condition numbers in the case of non-diagonalizable matrices, and discuss applications of that bound to matrix functions and spectrum perturbations. The main methodology presented in the paper is based on a combined usage of the recent norm estimates for matrix-valued functions with the traditional methods and results.


Introduction
This paper is a survey of the recent results of the author on perturbations of the eigenvalues and determinants of matrices.
Finding the eigenvalues of a matrix is not always an easy task. In many cases it is easier to calculate the eigenvalues of a nearby matrix and then to obtain the information about the eigenvalues of the original matrix.
The perturbation theory of matrices has been developed in the works of R. Bhatia To recall some basic results of the perturbation theory, which will be discussed below, let us introduce the notations.
Let C n be the n-dimensional complex Euclidean space with a scalar product (., .), the norm . = (., .) and unit matrix I. C n×n denotes the set of complex n × n-matrices. For an A ∈ C n×n , A * is the adjoint matrix, A −1 is the inverse one, A is the spectral norm: A = sup x∈C n , x =1 Ax , λ k (A) are the eigenvalues of A taken with their multiplicities, σ(A) is the spectrum, R λ (A) = (A − λI) −1 (λ ∈ σ(A)) is the resolvent, trace (A) is the trace, det A is the determinant, r s (A) is the spectral radius, and N p (A) := (trace (AA * ) p/2 ) 1/p (1 ≤ p < ∞) is the Schatten-von Neumann norm; in particular, N 2 (A) = A F is the Hilbert-Schmidt (Frobenius) norm.
| det A − detÃ| ≤ nM n−1 A −Ã (A,Ã ∈ C n×n ), (1) where M = max{ A , Ã }, cf. [1] (p. 107). The spectral norm is unitarily invariant, but often it is not easy to compute that norm, especially if the matrix depends on many parameters. In Section 4 below we present a bound for | det A − detÃ| in terms of the entries of matrices in the standard basis. That bound can be directly calculated. Moreover, under some conditions our bound is sharper than (1). Recall some definitions from matrix perturbation theory (see [2] (p. 167)).
The spectral variation ofÃ with respect to A is sv A (Ã) := max i min j |λ i − λ j |. The Hausdorff distance between the eigenvalues of A andÃ is hd(A,Ã) := max{sv A (Ã), svÃ(A)}.
Geometrically, the spectral variation has the following interpretation. If then σ(Ã) ⊂ ∪ n i=1 D i . In other words, the eigenvalues ofÃ lie in the union of disks of radius sv A (Ã) centered at the eigenvalues of A.
The Hausdorff distance hounds the spectral variation and is actually a metric. The matching distance bounds the Hausdorff distance and is also a metric. The "smallness" of the matching distance means that the eigenvalues of a matrix and its perturbation are "close" and they can be grouped into nearby pairs. In some cases bounds on the spectral variation or the Hausdorff distance can be converted into bound on the matching distance.
As it was mentioned, the calculations and estimating of the spectral norm is often a not easy task. Below we suggest bounds for the spectral variation and Hausdorff distance explicitly expressed via the entries of the considered matrices. In some cases our bounds are sharper than (3).
Furthermore, as is well-known, the Hilbert identity plays an important role in the perturbation theory. In Section 15, we suggest a new identity for resolvents and show that it refines the results derived with the help of the Hilbert identity, if the commutator AÃ −ÃA has a sufficiently small norm.
A few words about the contents of the paper. It consists of 17 Sections. In Section 2, we recall some classical results which are needed our proofs. In Section 3, we present norm estimates for resolvents of matrices which will be applied in the sequel.
In Sections 4 and 5, we derive the perturbation bound for determinants in terms of the entries of matrices and consider some its applications. Section 6 deals with perturbation bounds for determinants expressed via rather general norms.
Sections 7-10 are devoted to the spectral variations. Besides, the relevant bounds are obtained in terms of the departure from normality and via the entries of matrices.
Sections 11 and 12 deal with angular localization of matrices. The results of Section 12 are new. Section 13 is devoted to perturbations of diagonalizable matrices. Besides, we suggest a bound for the condition numbers. Besides, Corollary 14 is new.
As it was above mentioned, in Section 14 we generalize the Kahan result. In Sections 16 and 17, taking into account that any matrix having more than one eigenvalue is similar to a block-diagonal matrix, we obtain a bound for the condition numbers in the case of non-diagonalizable matrices, and discuss applications of that bound to matrix functions and spectrum perturbations. The material of Sections 16 and 17 is new.

Preliminaries
Recall the Schur theorem Section I.4.10.2 of [8], By that theorem there is an orthogonal normal (Schur's) basis {e k } n k=1 , in which A has the triangular representation a jk e j with a jk = (Ae k , e j ) (k = 1, . . . , n).
Schur's basis is not unique. We can write with a normal (diagonal) operator D defined by and a nilpotent operator V defined by a jk e j (k = 2, . . . , n), Ve 1 = 0.
Equality (5) is called the triangular representation of A; D and V are called the diagonal part and nilpotent part of A, respectively. Put (., e k )e k (j = 1, . . . , n), P 0 = 0.
{P k } n k=1 is called the maximal chain of the invariant projections of A. It has the properties 0 = P 0 C n ⊂ P 1 C n ⊂ · · · ⊂ P n C n = C n with dim (P k − P k−1 )C n = 1 and So A, V and D have the joint invariant subspaces. We can write where ∆P k = P k − P k−1 (k = 1, . . . , n). Let us recall also the famous Gerschgorin theorem [2] and Section III.2.2.1 of [8], which is an important tool for the analysis of the location of the eigenvalues. Theorem 1. The eigenvalues of A = (a jk ) ∈ C n×n lie in the union of the discs {z ∈ C : |z − a kk | ≤ n ∑ j=1,j =k |a jk |}, k = 1, . . . , n.
The Gerschgorin theorem implies the following inequality for the spectral radius:

Norm Estimates for Resolvents
The following quantity (the departure for normality) of A plays an essential role hereafter: By Lemma 3.1 from [9] g(A) = N 2 (V), where V is the nilpotent part of A (see equality (5)). Therefore, if A is a normal matrix, then g(A) = 0. The following relations are checked in Section 3.1 of [9]: and g(e iτ A + zI) = g(A) (z ∈ C, τ ∈ R). (8) By the inequality between the arithmetic and geometric means we have Hence, If A 1 ∈ C n×n and A 2 ∈ C n×n are commuting matrices, then g(A 1 + A 2 ) ≤ g(A 1 ) + g(A 2 ). Indeed, since A 1 and A 2 commute, they can have a joint basis of the triangular representation. So the nilpotent part of A 1 + A 2 is equal to V 1 + V 2 where V 1 and V 2 are the nilpotent parts of A 1 and A 2 , respectively. Therefore, We will need the following Theorem 2 (Theorem 3.1 of [9]). Let A ∈ C n×n . Then This Theorem sharp: if A is a normal matrix, then g(A) = 0 and we obtain R λ (A) = 1 ρ(A,λ) . Here and below we put 0 0 = 1. Let us recall an additional norm estimate for the resolvent, which is sharper than Theorem 2 but more cumbersome. To this end, for an integer n ≥ 2 introduce the numbers ψ n,k = ( n−1 k ) (n − 1) k (k = 1, . . . , n − 1) and γ n,0 = 1.
Theorem 4 (Theorem 3.4 of [9]). Let A ∈ C n×n . Then Let us point to an inequality between the resolvent and determinant.
Theorem 5. For any A ∈ C n×n and all regular λ of A one has For the proof see, for example Corollary 3.4 of [9].

Perturbation Bounds for Determinants in Terms of the Entries of Matrices
The following theorem is valid.
Theorem 6 (Reference [10]). Let A,Ã ∈ C n×n , {d k } be an arbitrary orthonormal basis in C n and q d = max j (A −Ã)d j . Then and, therefore, Proof. By the Hadamard inequality (see Section 2). Put It is not hard to check that Z(λ) is a polynomial in λ and ).
Obviously (A +Ã)d k , (A −Ã)d k (k = 1, . . . , n) are directly calculated. Below we also show that in the concrete situations Theorem 6 is sharper than (1) and enables us to establish sharp upper and lower bounds for the determinants of matrices that are "close" to triangular matrices.
Furthermore, making use of the inequality between the arithmetic and geometric means, from (11) we get Put A 1 = cA,Ã 1 = cÃ (c = const > 0). Then by the latter inequality Or Let us check that Indeed, the derivative of the function on the left-hand-side is Hence it follows that the infimum is reached at x = n − 1. This proves (14). So we can write | det A − detÃ| ≤ q d n n (n − 1) n−1 b n−1 .
We thus arrive at our next result. Corollary 1. Let A,Ã ∈ C n×n and {d k } be an arbitrary orthonormal basis in C n . Then we have

Perturbations of Triangular Matrices and Comparison with Inequality (1)
In this section, A = (a jk ) n j,k=1 ,Ã = (ã jk ) n j,k=1 , and {d k } is the standard basis. Clearly,

Now Theorem 6 implies
and, therefore, Furthermore, let A + be the upper triangular part of A, i.e., where a + jk = a jk if j ≤ k and a + jk = 0 for j > k. Then Clearly, Making use of Corollary 2, we arrive at our next result.

Corollary 3. One has
From this corollary we have Inequalities (15) and (17) are sharp: they are attained if A is triangular.
The following lemma taken from Lemma 3.3 of [10] gives us simple conditions, under which (11) is sharper than (1).
It should be noted that the determinants of diagonally dominant and double diagonally dominant matrices are very well explored, cf. [11][12][13][14]. At the same time the determinants of matrices "close" to triangular ones are investigated considerably less than the determinants of diagonally dominant matrices. About bounds for determinants of matrices close to the identity matrix see the papers [15].

Perturbation Bounds for Determinants in Terms of an Arbitrary Norm
Let A 0 be an arbitrary fixed matrix norm of A ∈ C n×n , i.e., the the function from C n×n into [0, ∞), defined by the usual relations: We need the following result.
Recall that N p (.) is the Schatten-von Neumann norm. Making use of the inequality between the arithmetic and geometric mean values, we obtain Due to the Weyl inequalities So in this case α n = 1 n n/p and γ n =η n,p , whereη n,p := n n(1−1/p) 2 n−1 (n − 1) n−1 .

Now Theorem 7 implies
Corollary 4. Let A, B ∈ C n×n . Then for any finite p ≥ 1, Note that Theorem 8.1.1 from the book [16] refines the Weyl inequality with the help of the self-commutator.
Furthermore, let i.e., W is the off-diagonal part of A: W = A − diag (a jj ). Then taking B = diag (a jj ) and making use of the previous corollary, we arrive at the following result.

Bounds for the Spectral Variations in Terms of the Departure from Normality
In this section, we estimate the spectral variation of two matrices in terms of the departure from normality g(A) introduced in Section 3. The results of the present section are based on the norm estimates for resolvents presented in Section 3 and the following technical lemma.

Lemma 2.
Let A andÃ be linear operators in C n and q := A −Ã . In addition, let where F(x) is a monotonically increasing continuous function of a non-negative variable x, such that F(0) = 0 and F(∞) = ∞. Then sv A (Ã) ≤ z(F, q), where z(F, q) is the unique positive root of the equation qF(1/z) = 1.
For the proof see Section 1.8 of [9]. Lemma 2 and Theorem 2 with If A is normal, then g(A) = 0, we have z n (A, q) = q and, therefore, Theorem 8 gives us the well-known inequality sv A (Ã) ≤ q, cf. [1,2]. Thus, Theorem 8 refines the Elsner inequality (3) if A is "close" to normal. Equation (21) can be written as To estimate z n (A, q) one can apply the well-known known bounds for the roots of polynomials. For instance, consider the algebraic equation with non-negative coefficients c j (j = 0, . . . , n − 1).

Lemma 3.
The unique positive rootẑ 0 of (23) satisfies the inequalitŷ Proof. Since all the coefficients of p(z) are non-negative, it does not decrease as z > 0 increases. (22), assuming that A is non-normal, i.e., g(A) = 0. Then we obtain the equation and applying Lemma 3 for the unique positive root x 0 of (24), we obtain But z n (A, q) = g(A)x 0 ; consequently, according to Theorem 8, we get Then Theorem 8 implies Replacing in Corollary 6 g(A) byĝ(Ã, A), we obtain the following result.

Corollary 7.
We have Now we are going to derive an estimate for the matching distance md(A,Ã) introduced in Section 1. To this end we need the following well-known result. (25), .
Making use of Theorem 9, we arrive at Since for a normal matrix A, g(A) = 0, Corollary 8 refines the Ostrowski-Elsner theorem mentioned in Section 1 for matrices close to normal ones.

A Bound for the Spectral Variation Via the Entries of Matrices
As mentioned above, the spectral norm is unitarily invariant, but the calculations and estimating of the spectral norm is often a not easy task, especially if the matrix depends on many parameters. In the paper [18], a bound for the spectral variation has been explicitly expressed via the entries of the considered matrices. In the paper [19], we have established a new bound via the entries. In the appropriate situations it considerably improves Elsner's inequality and the main result from [18]. In this section we present the main results from [19].
The proof of this theorem is presented in the next section. Simple calculations show that To illustrate Theorem 10 apply it with A = A + and A = A, taking into account that Now Theorem 10 implies.
Making use of (26), we arrive at Corollary 9. All the eigenvalues of A ∈ C n×n lie in the set ∪ n k=1 W k (A).

Proof of Theorem 10
In this section for the brevity put λ j (A) = λ j and λ j (Ã) =λ j . Lemma 4. Let A,Ã ∈ C n×n and {d k } be an arbitrary orthonormal basis in C n . Then for any eigenvalueλ j ofÃ we have min Proof. Due to Theorem 6, Hence, Since det(λ j I −Ã) = 0, (28) implies as claimed.

Angular Localization of the Eigenvalues of Perturbed Matrices
In this section we consider the following problem: let the eigenvalues of a matrix lie in a certain sector. In what sector do the eigenvalues of a perturbed matrix lie?
Not too many works are devoted to the angular localization of matrix spectra. The papers [20,21] should be mentioned. In these papers it is shown that the test to determine whether all eigenvalues of a complex matrix of order n lie in a certain sector can be replaced by an equivalent test to find whether all eigenvalues of a real matrix of order 4n lie in the left half-plane. Below we also recall the well-known results from Chapter 1, Exercise 32 of [22].
To the best of our knowledge, the problem just described of angular localization of the eigenvalues of perturbed matrices was not considered in the available literature, although it is important for various applications, cf. [22].
The results of this section are adopted from the paper [23].
Without loss of the generality, we assume that If this condition does not hold, instead of A we can consider perturbations of the matrix B = A + cI with a constant c > |β(A)|.
By the Lyapunov theorem, cf. Theorem I.5.1 of [22], condition (34) implies that there exists a positive definite Y ∈ C n×n , such that (YA) * + YA > 0. Define the angular Ycharacteristic τ(A, Y) of A by The set S(A, Y) := {z ∈ C : | arg z| ≤ τ(A, Y)} will be called the Y-spectral-sector of A. Let λ = re it (r > 0, 0 ≤ t < 2π) be an eigenvalue of A and d the corresponding eigenvector: Ad = λd. Then We, thus, get In Exercise 32, it is shown that the spectrum of A lies in the sector | arg z| ≤ dev(A). Since |(Ax, x)| ≤ Ax x , Lemma 5 refines the that inequality.
Furthermore, by the above mentioned Lyapunov theorem, there exists a positive definite X ∈ C n×n solving the Lyapunov equation Hence, Now we are in a position to formulate the main result of this section.
The proof of this theorem is based on the following lemma Lemma 6. Let A,Ã ∈ C n×n , condition (34) hold and X be a solution of (35). If, in addition, Proof. Put E =Ã − A. Then q = E and due to (35), with x = 1 we obtain In addition, according to (36) we arrive at the required result.
Proof of Theorem 11. Note that X is representable as Section 1.5 of [22]. Hence, we easily have X ≤ C J(A). Now the latter lemma proves the theorem.

Proof. By virtue of Example 3.2 from [9],
If A is normal, then g(A) = 0 and, taking 0 0 = 1 we haveĴ(A) = 1 β(A) . The latter lemma and Theorem 11.1 imply Corollary 10. Let A,Ã ∈ C n×n and the conditions (34) and qĴ(A) < 1 hold. Then Now consider the angular localization of the eigenvalues of matrices "close" to triangular ones. Let A + be the upper triangular part of A. i.e., A + = (a + jk ) n j,k=1 , where a + jk = a jk if j ≤ k and a + jk = 0 for j > k. To illustrate our results apply Corollary 10 with A instead of A and with A + instead of A.

Perturbations of Diagonalizable Matrices
An eigenvalue is said to be simple, if its geometric multiplicity is equal to one. In this section, we consider a matrix A whose all the eigenvalues are simple. As it is well known, in this case there is an invertible matrix T, such that whereD is a normal matrix. Besides, A is called a diagonalizable matrix. The condition number κ(A, T) := T T −1 is very important for various applications. We obtain a bound for the condition number and discuss applications of that bound to matrix functions and spectral variations.
If A ∈ C n×n (n ≥ 2) is diagonalizable, it can be written as whereQ k are one-dimensional eigen-projections. If f (z) is a scalar function defined on the spectrum of A, then f (A) is defined as be the interpolation Lagrange-Sylvester polynomial, such that r(λ k ) = f (λ k ). and cf. Section V.1 of [24]. From (40) it follows We thus arrive at

Lemma 8. Let A be diagonalizable and f (z) be a scalar function defined on the σ(A) for an
In particular, Inequality (41) and Lemma 7.1 imply.
Corollary 12. Let A,Ã ∈ C n×n and A be diagonalizable. Then Now we are going to estimate the condition number of A assuming that all the eigenvalues λ j of A are different: In other words the algebraic multiplicity of each eigenvalue is is equal to one. Recall that The proof of this theorem can be found in Theorem 6.1 of [9] and [25]. Theorem 12 is sharp: if A is normal, then g(A) = 0 and γ(A) = 1. Thus we obtain the equality κ(A, T) = 1. Lemma 8 and Theorem 12 immediately imply.

Corollary 13. Let condition (42) hold and f (z) be a scalar function defined on the σ(A) for an
Moreover, making use of Theorem 12 and Corollary 12, we arrive at the following result.

Corollary 14.
Let A,Ã ∈ C n×n and condition (42) hold. Then About additional inequalities for condition numbers via norms of the eigen-projections see [26,27]. About the functions of diagonalzable matrices see also [28].

Sums of Real Parts of Eigenvalues of Perturbed Matrices
The aim of the present section is to generalize the Kahan inequality (4). Again, put A R := (A + A * )/2 = Re A, A I := (A − A * )/2i = Im A and E =Ã − A. Let c m (m = 1, 2, . . . ) be a sequence of positive numbers defined by the recursive relation As it is proved in Corollary 1.3 of [29], Now we in a position to formulate and prove the main result of this section.
Here and below σ(A) denotes the spectrum of A. We haveÃ =D R + iD I +Ṽ and thus, the real and imaginary part of A arẽ respectively. Since A andD R are Hermitian, by the Mirsky inequality mentioned in the Introduction, we obtain Making use of Lemma 1.5 from [29], we get the inequality (see also Section 3.6 of [30,31]). In addition, by (48)Ṽ I =Ã I −D I and, therefore, Thanks to the above mentioned Weyl inequalities, Thus,

Now (50) implies the inequality
So by (49) we get the desired inequality The just proved theorem is sharp in the following sense: ifÃ is Hermitian, then N p (E I ) = 0 and inequality (47) becomes the Mirsky result, presented in Section 1.

Corollary 15.
Let a matrixÃ = (a jk ) n j,k=1 have the real diagonal entries. Let W be the off-diagonal part ofÃ: W =Ã − diag (a 11 , . . . , a nn ). Then for any p ∈ [2, ∞), and, therefore, Indeed, this result is due to the previous theorem with A = diag [a jj ]. Certainly, inequality (51) has a sense only if its right-hand side is positive. The case 1 ≤ p < 2 should be considered separately from the case p ≥ 2, since the relations between N p (Ṽ R ) and N p (Ṽ I ) similar to inequality (50) are unknown if p = 1, and we could not use the arguments of the proof of Theorem 13. The case 1 ≤ p < 2 is investigated in [32].

An Identity for Resolvents
Let A,Ã ∈ C n×n and E =Ã − A. The Hilbert identity for resolvents mentioned in Section 1 gives the following important result: if a λ ∈ C is regular for A and then λ is also regular forÃ. In this section we suggest a new identity for resolvents of matrices. It gives us new perturbation results which in appropriate situations improve condition (52). Put Z =ÃE − EA.

Theorem 14.
Let a λ ∈ C be regular for A andÃ. Then,

Proof. Put
Since the regular sets of operators are open, for t small enough, λ is a regular point of A t . By the previous lemma we get Hence, Thus, with the notation Take an integer m > c 0 (λ)/ E and put t k = k/m (k = 1, . . . , m). For m large enough, λ is a regular point of A t 1 and due to (54) we can write Hence, where γ = c 0 − E m . Due to inequality (55) we can assert that λ ∈ σ(A t 2 ). So in our arguments we can replace A t 1 by A t 2 and obtain the relations Therefore, λ ∈ σ(A t 3 ). Continuing this process for k = 4, . . . , m, we get λ ∈ σ(A t m ) = σ(Ã). Now (54) implies the required result.
Now the previous lemma yields the following result.
where C and B are commuting n × n-matrices. It is simple to check that Corollary 16 gives us the equality σ(T) = σ(T). At the same time, due to (52), if λ ∈ σ(T) we can assert that λ ∈ σ(T) only if T − T R λ (T) < 1.
If A is invertible, then due to Theorem 5,

Corollary 17. Suppose A is invertible, and
thenÃ is also invertible.
Recall that the quantity g(A) is introduced in Section 2. Theorems 2 and Corollary 16 imply our next result.

Corollary 18. If λ is regular for A and
then λ is regular forÃ.
The following theorem gives us the bound for the spectral variation via the identity for resolvents considered in this section.
Theorem 15. Let A andÃ be n × n matrices. Then sv A (Ã) ≤ x 1 , where x 1 is the unique positive root of the algebraic equation Proof. For any µ ∈ σ(A), due to Corollary 18 we have Hence, it follows that ρ(A, µ) ≤ x 1 , where x 1 is the unique positive root of the equation which is equivalent to (56). But sv A (Ã) = max j ρ(A, λ j (Ã)). This proves the theorem.
To estimate x 1 one can apply Lemma 13.
Let {e k } n k=1 be the corresponding orthonormal basis of the upper-triangular representation (the Schur basis). Denote (., e k )e k (i = 1, . . . , n); ∆Q k = (., e k )e k (k = 1, . . . , n); In addition, put A jk = ∆P j A∆P k (j = k) and A j = ∆P j A∆P j (j, k = 1, . . . , m). We can see that each P j is an orthogonal invariant projection of A and Besides, if µ j = 1, then A j = λ j ∆P j and ∆P j is one dimensional. If µ j > 1, then In the matrix form the blocks A j can be written as etc. Besides, each V j is a strictly upper-triangular (nilpotent) part of A j . So A j has the unique eigenvalue λ j of the algebraic multiplicity µ j : σ(A j ) = {λ j }. We, thus, have proved the following result.
Lemma 10. An arbitrary matrix A ∈ C n×n can be reduced by a unitary transform to the block triangular form (59) with A j = λ j ∆P j + V j ∈ C µ j ×µ j , where V j is either a nilpotent operator, or V j = 0. Besides, A j has the unique eigenvalue λ j of the algebraic multiplicity µ j .

Statement of the Main Result
Again, put Introduce, also, the notations .
It is not hard to check that d j ≤ 2 j . Now we are in a position to formulate the main result of this section.

Theorem 16.
Let an n × n-matrix A have m ≤ n (m ≥ 2) different eigenvalues λ j of the algebraic multiplicity µ j (j = 1, . . . , m). Then there are µ j × µ j -matrices A j each of which has a unique eigenvalue λ j , and an invertible matrix T, such that (58) holds with the block-diagonal matrix D = diag (A 1 , A 2 , . . . , A m ). Moreover, This theorem is proved in the next section. Theorem 16 is sharp: if A is normal, then g(A) = 0 and γ(A) = 1. Thus we obtain the equality κ T = 1.

Applications of Theorem 16
Let f (z) be a scalar function, regular on σ(A). Define f (A) by the usual way via the Cauchy integral [33]. Since A j are mutually orthogonal, we have Hence, (59) and (60) yield Corollary 19. Let A ∈ C n×n . Then there is an invertible matrix T, such that Due to Theorem 3.5 from the book [9] we have Take into account that g(A j ) ≤ g(A) (see Section 17). Now, making use of Theorem 16.2, we arrive at the following result.
Corollary 20. Let A ∈ C n×n . Then For example, we have About the recent results devoted to matrix-valued functions see for instance [9] and the references which are given therein. Now consider the resolvent. Then by (58) for |z| > max{ A , D } we have Extending this relation analytically to all regular z and taking into account that We get Corollary 21. Let A ∈ C n×n . Then there is an invertible matrix T, such that for any regular z of A.
But due to Theorem 3.2 from [9] we have where ρ(A, z) is the distance between z and the spectrum of A. Clearly, ρ(A j , z) ≥ ρ(A, z) (j = 1, . . . , m). Now Theorem 16 and (62) imply Furthermore, let A andÃ be complex n × n-matrices. Recall that sv A (Ã) is the spectral variation ofÃ with respect to A.

Proof of Theorem 16
Recall that P j are the orthogonal invariant projections defined in Section 16.1 and ∆P j = P j − P j−1 ; A jk and A j are also defined in Section 16.1. Put P k = I − P k , B k = P k AP k and C k = ∆P k AP k (k = 1, . . . , m − 1).
By Lemma 10 A j has the unique eigenvalue λ j and A is represented by (59). Represent B j and C j in the block-matrix form: Since B j is a block triangular matrix, it is not hard to see that cf. Lemma 6.2 of [9]. So due to Lemma 10, Under this condition, the equation has a unique solution X j : P j C n → ∆P j C n , e.g., Section VII.2 of [33].
So the matrix I − X j is inverse to I + X j . Thus, and (68) can be written as (58). We thus arrive at Corollary 24. Let an n × n-matrix A have m ≤ n (m ≥ 2) different eigenvalues λ j of the algebraic multiplicity µ j (j = 1, . . . , m). Then there are µ j × µ j -matrices A j each of which has a unique eigenvalue λ j and such that (58) holds with T defined by (70).
By the inequalities between the arithmetic and geometric means from (70)  (73) Proof of Theorem 16 Consider the Sylvester equation where B ∈ C n 1 ×n 1 ,B ∈ C n 2 ×n 2 and C ∈ C n 1 ×n 2 are given; X ∈ C n 1 ×n 2 should be found. Assume that the eigenvalues λ k (B) and λ j (B) of B andB, respectively, satisfy the condition. Then Equation (74) has a unique solution X, e.g., Section VII.2 of [33]. not mentioned in the article, plaese confirm and modifiy. Due to Corollary 5.8 of [9], the inequality whereĝ = max{g(B), g(B)}. Let us go back to Equation (66). In this case B = A j ,B = B j , C = C j , n 1 = µ j , n 2 = n j := dim P j C n , and due to (57), ρ 0 (A j , B j ) ≥ δ (j = 1, . . . , n). In addition, µ j +n j ≤ n. Now (75) implies whereĝ j = max{g(B j ), g(A j )}.
Recall that {e k } n k=1 denotes the Schur basis. So a jk e j with a jk = (Ae k , e j ) (j = 1, . . . , n).
We can write A = D A + V A ( σ(A) = σ(D A )) with a normal (diagonal) matrix D A defined by D A e j = a kk e k =λ j e k (k = 1, . . . , n) and a nilpotent (strictly upper-triangular) matrix V A defined by V A e k = a 1k e 1 + · · · + a k−1,k e k−1 (k = 2, . . . , n), V A e 1 = 0. D A and V A will be called the diagonal part and nilpotent part of A, respectively. It can be V A = 0, i.e., A is normal.
Besides, g(A) = V A F . In addition, the nilpotent part V j of A j is ∆P j V A ∆P j and the nilpotent part W j of B j is P j V A P j . So V j and W j are orthogonal, and Thus, from (76) it follows It can be directly checked that Thus, 2(m−1)
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.