Introduction

Many problems in science and engineering deal with eigenvalues and singular values of matrices. Here our main concern is perturbation bounds for matrix eigenvalues and singular values. In general, eigenvalue perturbation problems comprise the method to estimate the error when the eigenvalues of a perturbed matrix are approximated by the eigenvalues of an unperturbed matrix. Earlier, the authors working on perturbation theory have concentrated mainly on the bounds on the absolute differences between the approximate eigenvalue and true eigenvalue. However, there are practical situations when small eigenvalues should be determined to high relative accuracy. In this case, bounds on relative error are required. The main theme of this work is to obtain some relative perturbation bounds for eigenvalues of diagonalizable matrices. A result on relative perturbation bounds for singular value is also derived. Some techniques used in this work can be found in [1,2,3,4].

The following notations are used in this paper. \(M_n\) denotes the set of all matrices of order \(n \times n \) with complex entries and \(S_n\) is the set of all n! permutations of \(\{1,2, \ldots , n\}.\) \(A^t\) indicates the transpose of the matrix A. For a matrix \(A = (a_{ij}) \in M_n\) we use the following notations (see [5, 6]),

$$\begin{aligned} \Vert A\Vert _p = \left( \sum \limits _{i,j=1}^{n} |a_{ij}|^p \right) ^{\frac{1}{p}} \ \ \text{ for } \ p \ge 1, \end{aligned}$$
(1)

and

$$\begin{aligned} \Vert A\Vert _{q,p} = \left( \sum \limits _{j=1}^{n} \left( \sum \limits _{i=1}^{n} |a_{ij}|^q \right) ^{\frac{p}{q}} \right) ^{\frac{1}{p}} \ \ \text{ for } \ p \ge 1, \ \frac{1}{p}+ \frac{1}{q}= 1. \end{aligned}$$
(2)

For \(p=2,\) Eq. (1) gives the norm \(\Vert .\Vert _2\) which is known as Frobenius norm. The operator norm is denoted by \(\Vert .\Vert \).

One of the well-known theorem in the direction of absolute perturbation bound for matrix eigenvalues is the Hoffman–Weildant Theorem [7], which states that, if A and B are normal \(n \times n\) matrices and \(\{\lambda _1, \lambda _2, \ldots , \lambda _n\}\) and \(\{\mu _1, \mu _2, \ldots , \mu _n \}\) are their eigenvalues respectively then there exist a permutation \(\pi \in S_n\) such that,

$$\begin{aligned} \left( \sum \limits _{i=1}^{n} \Big | \lambda _i - \mu _{\pi (i)} \Big |^2 \right) ^{\frac{1}{2}} \le \Vert A-B\Vert _2. \end{aligned}$$
(3)

In 1998, Eisenstat and Ipsen [8] proved that a Hoffman–Weildant type absolute perturbation bound implies a relative bound for diagonalizable matrices. They proved that, if A and B both are diagonalizable matrices of order n and if A is non-singular then there exists a permutation \(\pi \in S_n\) such that,

$$\begin{aligned} \left( \sum \limits _{i=1}^{n} \Big | \frac{\lambda _i - \mu _{\pi (i)}}{\lambda _i} \Big |^2 \right) ^{\frac{1}{2}} \le \kappa (X) \kappa ({\tilde{X}}) \Vert A^{-1}(A-B)\Vert _2 \end{aligned}$$
(4)

where X and \({\tilde{X}}\) are the invertible matrices which diagonalize the matrices A and B respectively and \(\kappa (X)= \Vert X\Vert \Vert X^{-1}\Vert \) is the condition number of the matrix X. Furthermore, Li and Sun [9] obtained bounds by taking the matrix A as normal and non-singular and the matrix B arbitrary. Later Li and Chen [10] generalized the result obtained by Li and Sun [9] for diagonalizable matrices. Recently Chen [11] has proved a relative perturbation bound for singular values of \(n \times n\) matrices. If \(s_1 \ge s_2 \ge \cdots \ge s_n\) and \( {\tilde{s}}_1 \ge {\tilde{s}}_2 \ge \cdots \ge {\tilde{s}}_n\) are singular values of \(n \times n\) complex matrices A and B respectively, where A is non-singular, then he proved that, there exists a permutation \(\tau \in S_n\) such that,

$$\begin{aligned} \sqrt{\sum \limits _{i=1}^{n}\left( \frac{s_i - {\tilde{s}}_{\tau (i)}}{s_i}\right) ^2} \le \sqrt{\frac{\left||A^{-1}E\right||_2^2 + \left||EA^{-1}\right||_2^2}{2}}, \end{aligned}$$
(5)

where \(E = B-A.\)

The results mentioned earlier involve only Frobenius norm. Ikramov [12] partially generalized the the Hoffman–Weildant theorem with the norm \(\Vert .\Vert _p\) and obtained the following result. If A and B are hermitian matrices, then for \(1 < p \le 2\)

$$\begin{aligned} d_p(\lambda (A), \lambda (B)) \le \Vert A-B\Vert _p \end{aligned}$$
(6)

where \(d_p(\lambda (A), \lambda (B))\) is the “Holder distance” between the eigenvalues of A and B and it is given by

$$\begin{aligned} d_p(\lambda (A), \lambda (B)) = \min _{\pi \in S_n} \left( \sum \limits _{i=1}^{n} \Big | \lambda _i - \mu _{\pi (i)} \Big |^p \right) ^{\frac{1}{p}}. \end{aligned}$$
(7)

Recently Zhou et al. [4] generalized the above result for diagonalizable matrices.

Similar as Holder distance \(d_p(\lambda (A), \lambda (B))\) we define \(\tilde{d_p}(\lambda (A), \lambda (B))\) for relative perturbation as,

$$\begin{aligned} \tilde{d_p}(\lambda (A), \lambda (B)) = \min _{\pi \in S_n} \left( \sum \limits _{i=1}^{n} \Big | \frac{\lambda _i - \mu _{\pi (i)}}{\lambda _i} \Big |^p \right) ^{\frac{1}{p}}, ~~p \ge 1, \end{aligned}$$
(8)

where the matrix A is non-singular. A similar quantity \(\tilde{d_p}(s(A), s(B))\) can be defined for the singular values \(s(A) = \{s_1, \ldots , s_n\}\) of a non-singular matrix A and \(s(B) = \{{\tilde{s}}_1, \ldots , {\tilde{s}}_n\}\) of a matrix B. In this paper we investigated whether a bound similar as (6) holds for \(\tilde{d_p}(\lambda (A), \lambda (B))\) and \(\tilde{d_p}(s(A), s(B)).\) In “Relative Perturbation Bounds” section we will prove some upper and lower bounds for \(\tilde{d_p}(\lambda (A), \lambda (B))\) of diagonalizable matrices A and B. In “Relative Perturbation Bound for Singular Values” section we will derive an upper bound for \(\tilde{d_p}(s(A), s(B)).\)

Relative Perturbation Bounds

Before going to the main results we mention some basic relations which are useful. Let \(A=(a_{ij})\) and \(B=(b_{ij})\) are in \(M_n\) and let \(p \ge 1\) and q be such that \(\frac{1}{p}+\frac{1}{q}=1,\) then we have the following relations (see [6])

$$\begin{aligned}&\Vert AB\Vert _p \le \Vert A\Vert _p \Vert B\Vert _{q,p}, \end{aligned}$$
(9)
$$\begin{aligned}&\Vert AB\Vert _p \le \Vert A^t\Vert _{q,p} \Vert B\Vert _p. \end{aligned}$$
(10)

If B is non-singular, then

$$\begin{aligned} \frac{\Vert A\Vert _p}{\Vert B^{-1}\Vert _{q,p}} \le \Vert AB\Vert _p. \end{aligned}$$
(11)

If A is non-singular, then

$$\begin{aligned} \frac{\Vert B\Vert _p}{\Vert (A^{-1})^{t}\Vert _{q,p}} \le \Vert AB\Vert _p. \end{aligned}$$
(12)

For \(0 < p \le 2\)

$$\begin{aligned} \Vert AB\Vert _p \le \Vert A\Vert _p \Vert B\Vert _{p}. \end{aligned}$$
(13)

A square matrix of non-negative real numbers is called a doubly stochastic matrix if each row sum and each column sum is 1. A permutation matrix is a square matrix that has exactly one entry is 1 in each row and each column and 0 elsewhere. The Hadamard Product of two matrices with same dimension is defined by entrywise multiplication. If \(A=(a_{ij})\) and \(B=(b_{ij})\) are two matrices with same dimension then the Hadamard Product of A and B denoted by \(A \circ B\) and defined by

$$\begin{aligned} A \circ B = (a_{ij}b_{ij}). \end{aligned}$$

For any two \(n \times n\) real matrices A and B,  \(A \le _{e} B\) means \((B-A)\) is entrywise non-negative. Also if \(A=(a_{ij})\) be a \(n \times n\) matrix then we denote \(A^{|\circ |p} = (|a_{ij}|^p) \) for a real \(p > 0.\) We denote the singular values of A as \(s_1(A) \ge s_2(A) \ge \cdots \ge s_n(A).\) The next lemma involves entry wise inequality between two matrices.

Lemma 1

[13, Theorem 3.36] Let \(A \in M_n\) and let p, q be real numbers with \(0 < p \le 2\) and \(q \ge 2\). Then there exist two doubly stochastic matrices B, \(C \in M_n\) such that,

$$\begin{aligned} s_n^{p}(A) B \le _e A^{|\circ |p} \end{aligned}$$

and

$$\begin{aligned} A^{|\circ |q} \le _e s_1^{q}(A) C. \end{aligned}$$

We prove the following lemma which is useful in this section.

Lemma 2

Let \(Y=(y_{ij})\) be \(n \times n\) doubly stochastic matrix and \(M=(m_{ij}) \in M_n.\) Then there exist permutations \(\tau \) and \({\bar{\tau }}\) in \(S_n\) such that for \(p>0\),

$$\begin{aligned} \sum \limits _{i=1}^{n} |m_{i, \tau (i)} |^{p} \le \sum \limits _{i,j=1}^{n} |m_{ij} |^{p} y_{ij} \le \sum \limits _{i=1}^{n} |m_{i, {\bar{\tau }}(i)} |^{p}. \end{aligned}$$

Proof

Since Y is doubly stochastic, so by well-known Birkhoff’s Theorem it can be expressed as

$$\begin{aligned} Y= \sum \limits _{k=1}^{n!} \alpha _k P_k, \end{aligned}$$

where \(P_k\) are the permutation matrices and \(\sum \limits _{k=1}^{n!} \alpha _k = 1,\) \(\alpha _k \ge 0\). Let \(\tau _k\) are the corresponding permutations of the permutation matrices \(P_k\). For \(\mathrm{M=(m_{ij}) \in M_n}\)

$$\begin{aligned} M^{|\circ |p} \circ Y = \left( |m_{ij} |^{p} y_{ij} \right) . \end{aligned}$$

Let e denotes the \(n \times 1\) column vector whose components are all 1. Then

$$\begin{aligned} \sum \limits _{i,j=1}^{n} |m_{ij} |^{p} y_{ij} = e^t (M^{|\circ |p} \circ Y) e. \end{aligned}$$

Let

$$\begin{aligned}&e^t (M^{|\circ |p} \circ P) e = \min \{e^t (M^{|\circ |p} \circ P_k) e : 1 \le k \le n!\} \\&\text{ and } \ \ e^t (M^{|\circ |p} \circ {\bar{P}}) e = \max \{e^t (M^{|\circ |p} \circ P_k) e : 1 \le k \le n!\}. \end{aligned}$$

Also let \(\tau \) and \({\bar{\tau }}\) are the permutations corresponding P and \({\bar{P}}\) respectively. Therefore we have,

$$\begin{aligned} \sum \limits _{i,j=1}^{n} |m_{ij} |^{p} y_{ij} = e^t (M^{|\circ |p} \circ Y) e= & {} e^t \left( M^{|\circ |p} \circ \left( \sum \limits _{k=1}^{n!} \alpha _k P_k \right) \right) e\\= & {} e^t \left( \sum \limits _{k=1}^{n!} \alpha _k \left( M^{|\circ |p} \circ P_k\right) \right) e \\= & {} \sum \limits _{k=1}^{n!} \alpha _k \left( e^t(M^{|\circ |p} \circ P_k)e \right) \\\ge & {} \left( \min \limits _{1 \le k \le n!} e^t(M^{|\circ |p} \circ P_k)e \right) \sum \limits _{k=1}^{n!} \alpha _k\\= & {} e^t(M^{|\circ |p} \circ P)e. \end{aligned}$$

Since P is a permutation matrix and \(\tau \) be the corresponding permutation, so from above relation

$$\begin{aligned} \sum \limits _{i,j=1}^{n} |m_{ij} |^{p} y_{ij} \ge e^t(M^{|\circ |p} \circ P)e= \sum \limits _{i=1}^{n} |m_{i \tau (i)} |^{p}. \end{aligned}$$

Similarly for other part we have

$$\begin{aligned} \sum \limits _{i,j=1}^{n} |m_{ij} |^{p} y_{ij} = e^t (M^{|\circ |p} \circ Y) e= & {} e^t \left( M^{|\circ |p} \circ \left( \sum \limits _{k=1}^{n!} \alpha _k P_k \right) \right) e\\= & {} \sum \limits _{k=1}^{n!} \alpha _k \left( e^t(M^{|\circ |p} \circ P_k)e \right) \\\le & {} \left( \max \limits _{1 \le k \le n!} e^t(M^{|\circ |p} \circ P_k)e \right) \sum \limits _{k=1}^{n!} \alpha _k\\= & {} e^t(M^{|\circ |p} \circ {\bar{P}})e = \sum \limits _{i=1}^{n} |m_{i {\bar{\tau }}(i)} |^{p}. \end{aligned}$$

\(\square \)

Let us consider the matrices A and B in \(M_n\) with eigenvalues \(\{\alpha _1, \alpha _2, \ldots , \alpha _n\}\) and \(\{\beta _1, \beta _2, \ldots , \beta _n\}\) respectively and \({E} = A-B\) . Then there are invertible matrices P, Q and diagonal matrices \(D_1\), \(D_2\) such that,

$$\begin{aligned}&{A}= PD_1P^{-1} \end{aligned}$$
(14)
$$\begin{aligned}&{B}= QD_2 Q^{-1} \end{aligned}$$
(15)

where \(D_1\), \(D_2\) are diagonal matrices containing the eigenvalues of A, B respectively.

Theorem 1

If A, B are diagonalizable matrices and follow the above mentioned decompositions then there exists permutation \(\pi \) in \(S_n\) such that

$$\begin{aligned} \left[ \sum \limits _{i=1}^{n} \Big | \frac{\alpha _i - \beta _{\pi (i)}}{\alpha _i} \Big |^{p} \right] ^{\frac{1}{p}}\le & {} \Vert Q^{-1}\Vert \Vert P\Vert \Vert (P^{-1})^t\Vert _{q,p} \Vert Q\Vert _{q,p} \Vert A^{-1}E\Vert _p \end{aligned}$$

and

$$\begin{aligned} \left[ \sum \limits _{i=1}^{n} \Big | \frac{\alpha _i - \beta _{\pi (i)}}{\alpha _i} \Big |^{p} \right] ^{\frac{1}{p}}\le & {} \Vert Q^{-1}\Vert \Vert P\Vert \Vert P^{-1}\Vert _{p} \Vert Q\Vert _{p} \Vert A^{-1}E\Vert _p , \end{aligned}$$

where \(1< p \le 2\).

Proof

From Eqs. (14) and (15) we have

$$\begin{aligned} -A^{-1}{E}= & {} A^{-1}B - I = (PD_1^{-1}P^{-1})({Q}D_2 {Q}^{-1})- I. \end{aligned}$$

Multiplying \(P^{-1}\) from left and Q from right in the above relation we get,

$$\begin{aligned} -P^{-1}A^{-1}{E}Q= & {} D_1^{-1}P^{-1}{Q}D_2 - P^{-1}Q. \end{aligned}$$

Let \(\mathrm P^{-1}{Q} = Z = (z_{ij}).\) Then

$$\begin{aligned} -P^{-1}A^{-1}{E}Q= & {} D_1^{-1}ZD_2 - Z. \end{aligned}$$
(16)

Taking the norm \(\Vert . \Vert _p\) on both sides of the above equation where \(1 \le p <2,\) we get

$$\begin{aligned}&\Vert P^{-1}A^{-1}{E}Q\Vert _p^p = \sum \limits _{i,j=1}^{n} |z_{ij}|^{p} |\alpha _i^{-1} \beta _j - 1|^{p}. \end{aligned}$$

From Lemma 1 there exists a doubly stochastic matrix \(B=(b_{ij})\) such that for \(1< p \le 2\)

$$\begin{aligned} s_n^{p}(Z) b_{ij} \le |z_{ij}|^{p} \ \ \forall i,j. \end{aligned}$$

Therefore,

$$\begin{aligned}&\sum \limits _{i,j=1}^{n} b_{ij} |\alpha _i^{-1} \beta _j - 1|^{p} \le \Vert Z^{-1}\Vert ^{p} \Vert P^{-1}A^{-1}{E}Q\Vert _p^p. \end{aligned}$$

Again by Lemma 2 there exists a permutation \(\pi \in S_n\) such that,

$$\begin{aligned} \sum \limits _{i=1}^{n} |\alpha _i^{-1} \beta _{\pi (i)} - 1|^{p}\le & {} \Vert Z^{-1}\Vert ^{p} \Vert P^{-1}A^{-1}{E}Q\Vert _p^p. \nonumber \\ \text{ i.e., } \ \ \left[ \sum \limits _{i=1}^{n} |\alpha _i^{-1} \beta _{\pi (i)} - 1|^{p} \right] ^{\frac{1}{p}}\le & {} \Vert Q^{-1} P\Vert \Vert P^{-1}A^{-1}{E}Q\Vert _p. \end{aligned}$$
(17)

Using (9) and (10) we have

$$\begin{aligned} \left[ \sum \limits _{i=1}^{n} \left| \frac{\alpha _i - \beta _{\pi (i)}}{\alpha _i}\right| ^{p} \right] ^{\frac{1}{p}} \le \Vert Q^{-1}\Vert \Vert P\Vert \Vert (P^{-1})^t\Vert _{q,p} \Vert Q\Vert _{q,p} \Vert A^{-1}{E}\Vert _p. \end{aligned}$$

Also from (17) using (13) we have

$$\begin{aligned} \left[ \sum \limits _{i=1}^{n} \left| \frac{\alpha _i - \beta _{\pi (i)}}{\alpha _i}\right| ^{p} \right] ^{\frac{1}{p}} \le \Vert Q^{-1}\Vert \Vert P\Vert \Vert P^{-1}\Vert _{p} \Vert Q\Vert _{p} \Vert A^{-1}{E}\Vert _p. \end{aligned}$$

\(\square \)

Theorem 2

Under the same assumption of Theorem 1 there is permutation \(\sigma \) in \(S_n\) such that

$$\begin{aligned} \left[ \sum \limits _{i=1}^{n} \left| \frac{\alpha _i - \beta _{\sigma (i)}}{\alpha _i}\right| ^{q} \right] ^{\frac{1}{q}}\ge & {} \frac{\Vert A^{-1}{E}\Vert _q}{\Vert P^{-1}\Vert \Vert Q\Vert \Vert P^t\Vert _{p,q} \Vert {Q}^{-1}\Vert _{p,q}}. \end{aligned}$$

where \(q \ge 2\).

Proof

For \(2 \le q,\) taking norm \(\Vert .\Vert _q\) on both sides of (16) we have,

$$\begin{aligned}&\Vert P^{-1}A^{-1}{E}Q\Vert _q^q = \sum \limits _{i,j=1}^{n} |z_{ij}|^{q} |\alpha _i^{-1} \beta _j - 1|^{q}. \end{aligned}$$

From Lemma 1 there exists a doubly stochastic matrix \(C=(c_{ij})\) such that

$$\begin{aligned} |z_{ij}|^{q} \le s_1^{q}(Z) c_{ij} \ \ \forall i,j , \end{aligned}$$

where \(Z=P^{-1}Q.\) Therefore,

$$\begin{aligned} \sum \limits _{i,j=1}^{n} c_{ij} |\alpha _i^{-1} \beta _j - 1|^{q}\ge & {} \left( \Vert Z\Vert ^{q} \right) ^{-1} \Vert P^{-1}A^{-1}{E}Q\Vert _q^q. \end{aligned}$$

By Lemma 2 there exists a permutation \(\sigma \in S_n\) such that,

$$\begin{aligned} \sum \limits _{i,j=1}^{n} |\alpha _i^{-1} \beta _{\sigma (i)} - 1|^{q}\ge & {} \frac{\Vert P^{-1}A^{-1}{E}Q\Vert _q^q}{ \Vert Z\Vert ^{q}}. \end{aligned}$$

Using (11) and (12) we get

$$\begin{aligned} \left[ \sum \limits _{i=1}^{n} \left| \frac{\alpha _i - \beta _{\sigma (i)}}{\alpha _i}\right| ^{q} \right] ^{\frac{1}{q}}\ge & {} \frac{\Vert P^{-1}A^{-1}{E}Q\Vert _q}{ \Vert P^{-1}Q\Vert }\\\ge & {} \frac{\Vert A^{-1}{E}\Vert _q}{\Vert P^{-1}\Vert \Vert Q\Vert \Vert P^t\Vert _{p,q} \Vert {Q}^{-1}\Vert _{p,q}}. \end{aligned}$$

\(\square \)

Relative Perturbation Bound for Singular Values

Let \(A, B \in M_n\) where A is non-singular, the singular value decompositions (SVD)

$$\begin{aligned} A= U \varSigma V^* \quad \text{ and } \quad B= U_1 {\tilde{\varSigma }} V^*_1 \end{aligned}$$

respectively, where \(U, V^{*}, U_1, V^*_1\) are \(n \times n\) unitary matrices, and

$$\begin{aligned} \varSigma = diag(s_1, \ldots , s_n) \quad \text{ and } \quad {\tilde{\varSigma }} = diag({\tilde{s}}_1, \ldots , {\tilde{s}}_n), \end{aligned}$$

and \(s_1 \ge s_2 \ge \cdots \ge s_n\) and \( {\tilde{s}}_1 \ge {\tilde{s}}_2 \ge \cdots \ge {\tilde{s}}_n\) are singular values of A and B respectively. In the next Theorem we use the following relation,

$$\begin{aligned} (a^p + b^p) \ge 2^{1-p} (a+b)^p, a,b \ge 0, \ \ p \ge 1. \end{aligned}$$

Theorem 3

Let \(A, B \in M_n\) with A is non-singular, follow the above mentioned singular value decomposition. Then there exists permutation \(\pi \) of \(S_n\) such that for \(p>1\),

$$\begin{aligned} \left[ \sum \limits _{i=1}^{n} \Big | \frac{s_i - {\tilde{s}}_{\pi (i)}}{s_i}\Big |^p \right] ^{\frac{1}{p}}\le \left[ 2^{p-1} ({\Vert V^* A^{-1}E V_1\Vert _{p}^{p} + \Vert U^* (E A^{-1})^* U_1 \Vert ^p_p}) \right] ^{\frac{1}{p}}, \end{aligned}$$

where \(E= B-A.\)

Proof

Given \(A= U \varSigma V^*\) and \(B= U_1 {\tilde{\varSigma }} V^*_1,\) where \(U, V, U_1, V_1\) are \(n \times n\) unitary matrices. Since A is non-singular so,

$$\begin{aligned} A^{-1}E= & {} A^{-1}(A+E) - I \\= & {} (V \varSigma ^{-1} U^*) (U_1 {\tilde{\varSigma }} V^*_1) - I , \\ \Rightarrow V^* A^{-1}E V_1= & {} \varSigma ^{-1} U^* U_1 {\tilde{\varSigma }} - V^*V_1\\= & {} \varSigma ^{-1} P {\tilde{\varSigma }} - Q, \end{aligned}$$

where \(U^* U_1= P=(p_{ij})\) and \(V^*V_1 = Q = (q_{ij}).\) Therefore,

$$\begin{aligned} \Vert V^* A^{-1}E V_1\Vert _{p}^{p} = \sum \limits _{i,j=1}^{n} \Big | \frac{\tilde{s_j}}{s_i}p_{ij} - q_{ij} \Big |^{p}. \end{aligned}$$
(18)

Also

$$\begin{aligned} EA^{-1}= & {} (A+E)A^{-1} - I \\= & {} (U_1 {\tilde{\varSigma }} V^*_1) (V \varSigma ^{-1} U^*) - I. \end{aligned}$$

In addition

$$\begin{aligned} (E A^{-1})^*= & {} U \varSigma ^{-1} V^* V_1 {\tilde{\varSigma }} U_1^* - I\\ \Rightarrow U^* (E A^{-1})^* U_1= & {} \varSigma ^{-1} Q {\tilde{\varSigma }} - P. \end{aligned}$$

Therefore

$$\begin{aligned} \Vert U^* (E A^{-1})^* U_1 \Vert ^p_p = \sum \limits _{i,j=1}^{n} \Big | \frac{\tilde{s_j}}{s_i}q_{ij} - p_{ij} \Big |^{p}. \end{aligned}$$
(19)

Adding the equations (18) and (19) we get,

$$\begin{aligned} \Vert V^* A^{-1}E V_1\Vert _{p}^{p} + \Vert U^* (E A^{-1})^* U_1 \Vert ^p_p= & {} \sum \limits _{i,j=1}^{n} \left[ \Big | \frac{\tilde{s_j}}{s_i}p_{ij} - q_{ij} \Big |^{p} + \Big | \frac{\tilde{s_j}}{s_i}q_{ij} - p_{ij} \Big |^{p} \right] \\\ge & {} 2^{1-p} \sum \limits _{i,j=1}^{n} \left[ \Big | \frac{\tilde{s_j}}{s_i}p_{ij} - q_{ij} \Big | + \Big | \frac{\tilde{s_j}}{s_i}q_{ij} - p_{ij} \Big | \right] ^{p}\\\ge & {} 2^{1-p} \sum \limits _{i,j=1}^{n} \left[ \Big | \frac{\tilde{s_j}}{s_i}|p_{ij}| - |q_{ij}| \Big | + \Big | \frac{\tilde{s_j}}{s_i}|q_{ij}| - |p_{ij}| \Big | \right] ^{p}\\\ge & {} 2^{1-p} \sum \limits _{i,j=1}^{n} \left[ \Big | \frac{\tilde{s_j}}{s_i}|p_{ij}| - |q_{ij}| + \frac{\tilde{s_j}}{s_i}|q_{ij}| - |p_{ij}| \Big | \right] ^{p}\\= & {} 2^{1-p} \sum \limits _{i,j=1}^{n} \left[ \Big | \frac{\tilde{s_j}}{s_i}(|p_{ij}|+|q_{ij}|) -(|p_{ij}|+|q_{ij}|) \Big | \right] ^{p}\\= & {} 2^{1-p} \sum \limits _{i,j=1}^{n} \left[ \Big | \frac{\tilde{s_j}}{s_i}-1\Big |^p (|p_{ij}|+|q_{ij}|)^p \right] \\\ge & {} 2^{1-p} \sum \limits _{i,j=1}^{n} \left[ \Big | \frac{\tilde{s_j}}{s_i}-1\Big |^p |p_{ij}|^p \right] \end{aligned}$$

From Lemma 1 there exists double stochastic matrix \(B=(b_{ij})\) such that,

$$\begin{aligned} \Vert V^* A^{-1}E V_1\Vert _{p}^{p} + \Vert U^* (E A^{-1})^* U_1 \Vert ^p_p \ge 2^{1-p} s_n^p({P}) \sum \limits _{i,j=1}^{n} \left[ \Big | \frac{\tilde{s_j}}{s_i}-1\Big |^p b_{ij} \right] . \end{aligned}$$

Again from Lemma 2 there exists a permutation \(\pi \) of \(S_n\) such that,

$$\begin{aligned} \Vert V^* A^{-1}E V_1\Vert _{p}^{p} + \Vert U^* (E A^{-1})^* U_1 \Vert ^p_p \ge 2^{1-p} \sum \limits _{i=1}^{n} \left[ \Big | \frac{{\tilde{s}}_{\pi (i)}}{s_i}-1\Big |^p \right] , \end{aligned}$$

which yields the desired result. \(\square \)

Remark 1

The results on relative perturbation bounds obtained in “Relative Perturbation Bounds” section generalize Corollary 5.2 of [8] and the result for singular values, proved in “Relative Perturbation Bound for Singular Values” section generalize Theorem 2.2 of [11].

As we have mentioned previously, relative error bounds are required when there are eigenvalues of small magnitude. There are practical situations where small eigenvalues have physical meaning and such situations include computing modes of vibration in a finite element context, and computing energy levels in quantum mechanical systems [14]. Also, relative perturbation bounds for eigenvalues and singular values have an applications in developing high accuracy numerical methods for eigenvalues and singular value problems [15, 16].