1 Introduction

The numerical analysis for an eigenpair of a matrix plays an important role in science engineering computing, physical science et al. Usually, the numerical analysis for eigenpairs contains two aspects:

  • algorithms for computing the eigenvalue and the eigenvector;

  • (absolute and relative) perturbation analysis for additive perturbations and multiplicative perturbations, which can lead to the condition number for computing the eigenvalue and the eigenvector.

There have been many significant results for the perturbation analysis (see, e.g., [20]). The classical bounds for the eigenpair perturbation are the Hoffman–Wielandt theorem for eigenvalues [6] and the \(\sin \varTheta \) theorem for eigenspaces [3], respectively. Recently, some new perturbation bounds for eigenpairs have been obtained; see [1, 2, 4, 7, 8, 12,13,14, 16, 19]. Some combined perturbations for matrix decompositions were established; see [2, 14, 16, 17]. In particular, some combined perturbation bounds for eigenpairs of Hermitian matrices were presented; see [14, 16]. In order to give combined bounds for eigenpairs of diagonalizable matrices, we first introduce some notations.

Let \(\mathbf {C}^{m\times n}\) be the set of \(m\times n\) complex matrices, and let \(\langle n\rangle =\{1,2,\ldots , n\}.\) By \(A^{*}\) and I we denote the conjugate transpose of a matrix A and the identity matrix, respectively. The Frobenius norm, the spectral norm, the minimum and maximum singular value of a matrix are denoted by \(\Vert \cdot \Vert _{F}\), \(\Vert \cdot \Vert _{2}\), \(\sigma _{\text {min}}(\cdot )\) and \(\sigma _{\text {max}}(\cdot )\), respectively.

Let both A and its perturbed matrix \({\widetilde{A}}\) be \(n\times n\) diagonalizable matrices with the following eigendecompositions:

$$\begin{aligned} AX=X\varLambda \equiv \left( \begin{array}{ll} X_{1}&X_{2} \end{array} \right) \left( \begin{array}{ll} \varLambda _{1} &{} 0 \\ 0 &{} \varLambda _{2} \end{array} \right) \quad \hbox {and}\quad {\widetilde{A}}{\widetilde{X}}={\widetilde{X}}\widetilde{ \varLambda }\equiv \left( \begin{array}{ll} {\widetilde{X}}_{1}&{\widetilde{X}}_{2} \end{array} \right) \left( \begin{array}{ll} {\widetilde{\varLambda }}_{1} &{} 0 \\ 0 &{} {\widetilde{\varLambda }}_{2} \end{array} \right) , \end{aligned}$$
(1.1)

where X and \({\widetilde{X}} \in \mathbf {C}^{n\times n}\) are nonsingular, \( X_{1}\) and \({\widetilde{X}}_{1} \in \mathbf {C}^{n\times r}\), \(1\le r \le n\),

$$\begin{aligned} \varLambda _{1}= & {} \text { diag}(\lambda _{1},\lambda _{2},\ldots ,\lambda _{r}), \quad \varLambda _{2}=\text { diag}(\lambda _{r+1},\lambda _{r+2},\ldots ,\lambda _{n}), \end{aligned}$$
(1.2)
$$\begin{aligned} \overset{\sim }{\varLambda }_{1}= & {} \text { diag}(\overset{\sim }{\lambda }_{1},\overset{ \sim }{\lambda }_{2},\ldots ,\overset{\sim }{\lambda }_{r}), \quad \overset{\sim }{\varLambda }_{2}=\text { diag}(\overset{\sim }{\lambda }_{r+1},\overset{ \sim }{\lambda }_{r+2},\ldots ,\overset{\sim }{\lambda }_{n}), \end{aligned}$$
(1.3)

and \(\lambda _{i}\) and \(\overset{\sim }{\lambda }_{j}\) may be complex. Partition

$$\begin{aligned} X^{-1}=\left( \begin{array}{c} Y_{1}^{*} \\ Y_{2}^{*} \end{array} \right) \quad \hbox {and}\quad {\widetilde{X}}^{-1}=\left( \begin{array}{c} {\widetilde{Y}}_{1}^{*} \\ {\widetilde{Y}}_{2}^{*} \end{array} \right) \,, \end{aligned}$$
(1.4)

where \(Y_{1}, {\widetilde{Y}}_{1}\in \mathbf {C}^{n\times r}\). Let

$$\begin{aligned} \delta _{ij}^{(l,k)}=\min \limits _{\lambda \in \lambda (\varLambda _{i}), {\widetilde{\lambda }}\in \lambda ({\widetilde{\varLambda }}_{j})}\frac{|\lambda - {\widetilde{\lambda }}|}{ |\lambda |^{l}|{\widetilde{\lambda }}|^{k}},\quad i,j=1,2. \end{aligned}$$
(1.5)

For simplicity, we always use the notation \(\delta _{ij}=\delta _{ij}^{(0,0)}\). Another relative gap is given by

$$\begin{aligned} \rho _{ij}^{ (l,k)}=\min \limits _{\lambda \in \lambda (\wedge _{i}),\overset{\sim }{\lambda }\in \lambda (\overset{\sim }{\wedge }_{j})}\frac{\mid \lambda -{\widetilde{\lambda }}\mid }{|\lambda |^{l}|{\widetilde{\lambda }}|^{k} \sqrt{\smash [b]{\mid {\widetilde{\lambda }}\mid ^{2}+\mid \lambda \mid ^{2}}}},\; i,j=1,2, \end{aligned}$$
(1.6)

where k and \(\ l\) are nonnegative real numbers.

Let \(X_1\) and \({\widetilde{X}}_1\) \(\in \mathbf {C}^{n \times r}( n\ge r)\) have full column rank r. Then the angle matrix \(\varTheta (X_1,{\widetilde{X}}_1)\) between \(X_1\) and \({\widetilde{X}}_1\) is defined by [20]:

$$\begin{aligned} \varTheta (X_1,{\widetilde{X}}_1)=\arccos ((X_1^{*}X_1)^{-\frac{1}{2}}X_1^{*} {\widetilde{X}}_1({\widetilde{X}}_1^{*}{\widetilde{X}}_1)^{-1}{\widetilde{X}}_1^{*}X_1(X^{*}X_1)^{-\frac{1}{2}})^{\frac{1}{2}}. \end{aligned}$$

In particular, if both \(X_{1}\) and \({\widetilde{X}}_{1}\) have orthonormal columns, then for any unitarily invariant norm \(\Vert \cdot \Vert \) we have

$$\begin{aligned} \Vert \sin \varTheta (X_{1},{\widetilde{X}}_{1})\Vert =\Vert X_{1}^{*} {\widetilde{X}}_{2}\Vert =\Vert {\widetilde{X}}_{1}^{*}X_{2}\Vert , \end{aligned}$$

where \((X_{1},X_{2})\) and \(({\widetilde{X}}_{1}, {\widetilde{X}}_{2})\) are \(n\times n\) unitary matrices.

Let A and \({\widetilde{A}}=A+\varDelta A\) have the decomposition (1.1)–(1.3). Here we consider the relative perturbation for the eigenpair.

For the relative perturbation of eigenvalues and eigenspaces, Ipsen presented the following general relative bounds

$$\begin{aligned} \sqrt{\sum _{i=1}^{n}\left| {\widetilde{\lambda }}_{i}^{-k}\lambda _{\tau (i)}^{1-l}- {\widetilde{\lambda }}_{i}^{1-k}\lambda _{\tau (i)}^{-l}\right| ^{2}}\le \kappa \left( {\widetilde{X}}\right) \kappa (X) \Vert {\widetilde{A}}^{-k}\varDelta AA^{-l}\Vert _{F} \end{aligned}$$
(1.7)

and

$$\begin{aligned} \Vert \sin \varTheta ({\widetilde{X}}_{1},X_{1})\Vert _{F}\le \kappa (\widetilde{ Y}_{2})\kappa (X_{1})\frac{\Vert {\widetilde{A}}^{-k}\varDelta AA^{-l}\Vert _{F}}{ \delta _{12}^{(l,k)}}, \end{aligned}$$
(1.8)

respectively, where \(\kappa (B)=\Vert B\Vert _{2}\Vert B^{-1}\Vert _{2}\) for any nonsingular matrix B or \(\kappa (B)=\Vert B\Vert _{2}\Vert B^{\dag }\Vert _{2}\) for any matrix B (see Theorem 6.1 and Corollary 3.3 of [8]).

The idea of this paper is to combine (1.7) and (1.8) together into one formula, from which one may deduce some classical perturbation bounds for eigenvalues and eigenspaces, respectively.

The rest of this paper is organized as follows. In Sect. 2, we present the combined bound for eigenvalues and eigenspaces in the additive perturbation case. In Sect. 3, we consider the multiplicative perturbation case, and get the combined bound for the multiplicative perturbation. In Sect. 4, we give some numerical example to show the theoretical results. Some concluding remarks are given in the final section.

2 Relative bounds for additive perturbation

In this section we will get a relative combined perturbation bound for the eigenpair. First of all, we give some lemmas which will be used in this section.

The following Lemma 2.1 was implicitly hidden in the presentation of Sun [18], but formally stated in Lemma 2.2 of Li [9], and also can be found in [13] and [15].

Lemma 2.1

Suppose that \(X=(X_{1},X_{2})\in \mathbf {C}^{n\times n}\) is a nonsingular matrix, where \(X_{1}\) \( \in \mathbf {C}^{n\times m}\), and its inverse has the block form (1.4). Then for the 2-norm or Frobenius norm \(\Vert \cdot \Vert \) and any full column matrix \({\widetilde{X}}_{1}\) \(\in \mathbf {C}^{n\times m},\)

$$\begin{aligned} \Vert \sin \varTheta (X_{1,}{\widetilde{X}}_{1})\Vert \le \Vert Y_{2}^{\dagger }\Vert _{2}\Vert {\widetilde{X}}_{1}^{\dagger }\Vert _{2}\Vert Y_{2}^{*} {\widetilde{X}}_{1}\Vert , \end{aligned}$$

where by \(M^{\dagger }\) we denote the Moore–Penrose inverse of a matrix M.

The following lemma is Theorem 3.2 in [5], which is further generalized to more general case in Proposition 3.1 of [11]; see also the last line of [5] for more details.

Lemma 2.2

[5] Let \(T\in \mathbf {C}^{n \times n}\) and \(\varLambda _{i}=diag(\lambda _{1}^{(i)},\ldots ,\lambda _{n}^{(i)})\in \mathbf {C}^{n\times n},\;i=1,2,3,4\). Then there exists a permutation \(\tau \) of \(\langle n\rangle \) such that

$$\begin{aligned} \sigma _{\text {min}}^{2}(T) \sum _{i=1}^{n}\left| \lambda _{i}^{(1)}\lambda _{\tau (i)}^{(2)}-\lambda _{i}^{(3)}\lambda _{\tau (i)}^{(4)}\right| ^{2}\le \Vert \varLambda _{1}T\varLambda _{2}-\varLambda _{3}T\varLambda _{4}\Vert _{F}^{2}. \end{aligned}$$

Now we present a combined perturbation bound for the relative measure.

Theorem 2.3

Let A and its perturbed matrix \( {\widetilde{A}}\) be two \(n\times n\) nonsingular diagonalizable matrices with the eigendecompositions (1.1)–(1.3). Then there exists a permutation \(\tau \) of \(\langle r\rangle \) such that

$$\begin{aligned}&\frac{\left( \delta _{12}^{(l,k)}\right) ^{2}}{\Vert {\widetilde{Y}} _{2}^{\dag }\Vert _{2}^{2}\Vert X_{1}^{\dag }\Vert _{2}^{2}}\Vert \sin \varTheta ({\widetilde{X}}_{1},X_{1})\Vert _{F}^{2}+\sigma _{\text {min}}^{2}(\widetilde{Y }_{1}^{*}X_{1})\sum _{i=1}^{r}|{\widetilde{\lambda }}_{i}^{-k}\lambda _{\tau (i)}^{1-l}-{\widetilde{\lambda }}_{i}^{1-k}\lambda _{\tau (i)}^{-l}|^{2} \nonumber \\&\quad \le \Vert {\widetilde{X}}^{-1}{\widetilde{A}}^{-k}\varDelta AA^{-l}X_{1}\Vert _{F}^{2}, \end{aligned}$$
(2.1)

where \(\delta _{12}^{(l,k)}\) is given by (1.5).

Proof

It is easy to check

$$\begin{aligned} {\widetilde{A}}^{-k}(A-{\widetilde{A}})A^{-l}={\widetilde{A}}^{-k}A^{1-l}- {\widetilde{A}}^{1-k}A^{-l}, \end{aligned}$$

i.e.,

$$\begin{aligned} {\widetilde{X}}{\widetilde{\varLambda }}^{-k}{\widetilde{X}}^{-1}X\varLambda ^{1-l}X^{-1}-{\widetilde{X}}{\widetilde{\varLambda }}^{1-k}{\widetilde{X}} ^{-1}X\varLambda ^{-l}X^{-1}={\widetilde{A}}^{-k}(A-{\widetilde{A}})A^{-l}. \end{aligned}$$

Hence,

$$\begin{aligned} {\widetilde{\varLambda }}^{-k}{\widetilde{X}}^{-1}X\varLambda ^{1-l}-\widetilde{ \varLambda }^{1-k}{\widetilde{X}}^{-1}{X}\varLambda ^{-l}={\widetilde{X}}^{-1}\widetilde{ A}^{-k}(A-{\widetilde{A}})A^{-l}X. \end{aligned}$$

Let \(\varDelta A=A-{\widetilde{A}}.\) By (1.4) we have

$$\begin{aligned} {\widetilde{X}}^{-1}X_{1}=\left( \begin{array}{c} {\widetilde{Y}}_{1}^{*}X_{1} \\ {\widetilde{Y}}_{2}^{*}X_{1} \end{array} \right) . \end{aligned}$$

Then

$$\begin{aligned} \left( \begin{array}{c} {\widetilde{\varLambda }}_{1}^{-k}{\widetilde{Y}}_{1}^{*}X_{1}\varLambda _{1}^{1-l}-{\widetilde{\varLambda }}_{1}^{1-k}{\widetilde{Y}}_{1}^{*}X_{1}\varLambda _{1}^{-l} \\ {\widetilde{\varLambda }}_{2}^{-k}{\widetilde{Y}}_{2}^{*}X_{1}\varLambda _{1}^{1-l}-{\widetilde{\varLambda }}_{2}^{1-k}{\widetilde{Y}}_{2}^{*}X_{1}\varLambda _{1}^{-l} \end{array} \right) ={\widetilde{X}}^{-1}{\widetilde{A}}^{-k}\varDelta AA^{-l}X_{1}. \end{aligned}$$
(2.2)

Taking the Frobenius norm on both sides of (2.2) gives

$$\begin{aligned}&\left\| {\widetilde{\varLambda }}_{2}^{-k}{\widetilde{Y}}_{2}^{*}X_{1}\varLambda _{1}^{1-l}-{\widetilde{\varLambda }}_{2}^{1-k}{\widetilde{Y}}_{2}^{*}X_{1}\varLambda _{1}^{-l}\right\| _{F}^{2}+\left\| {\widetilde{\varLambda }}_{1}^{-k} {\widetilde{Y}}_{1}^{*}X_{1}\varLambda _{1}^{1-l}-{\widetilde{\varLambda }} _{1}^{1-k}{\widetilde{Y}}_{1}^{*}X_{1}\varLambda _{1}^{-l}\right\| _{F}^{2} \nonumber \\&\quad =\left\| {\widetilde{X}}^{-1}{\widetilde{A}}^{-k}\varDelta AA^{-l}X_{1}\right\| _{F}^{2} . \end{aligned}$$
(2.3)

It is easy to see that

$$\begin{aligned}&\left| \left( {\widetilde{\varLambda }}_{2}^{-k}{\widetilde{Y}}_{2}^{*}X_{1}\varLambda _{1}^{1-l}-{\widetilde{\varLambda }}_{2}^{1-k}{\widetilde{Y}}_{2}^{*}X_{1}\varLambda _{1}^{-l}\right) _{ij}\right| ^{2} \\&\quad =\left( \lambda _{j}^{1-l}{\widetilde{\lambda }} _{i}^{-k}-\lambda _{j}^{-l}{\widetilde{\lambda }}_{i}^{1-k}\right) ^{2}\left| \left( {\widetilde{Y}} _{2}^{*}X_{1}\right) _{ij}\right| ^{2}\ge \left( \delta _{12}^{(l,k)}\right) ^{2}\left| ({\widetilde{Y}}_{2}^{*}X_{1})_{ij}\right| ^{2}. \end{aligned}$$

Thus

$$\begin{aligned} \left( \delta _{12}^{(l,k)}\right) ^{2}\left\| {\widetilde{Y}}_{2}^{*}X_{1}\right\| _{F}^{2}\le \left\| {\widetilde{\varLambda }}_{2}^{-k}{\widetilde{Y}} _{2}^{*}X_{1}\varLambda _{1}^{1-l}-{\widetilde{\varLambda }}_{2}^{1-k}\widetilde{ Y}_{2}^{*}X_{1}\varLambda _{1}^{-l}\right\| _{F}^{2}. \end{aligned}$$
(2.4)

By Lemma 2.1,

$$\begin{aligned} \Vert \sin \varTheta ({\widetilde{X}}_{1},X_{1})\Vert _{F}\le \Vert {\widetilde{Y}} _{2}^{\dag }\Vert _{2}\Vert X_{1}^{\dag }\Vert _{2}\Vert {\widetilde{Y}} _{2}^{*}X_{1}\Vert _{F}, \end{aligned}$$

which together with (2.4) gives

$$\begin{aligned} \frac{\left( \delta _{12}^{(l,k)}\right) ^{2}}{\Vert {\widetilde{Y}}_{2}^{\dag }\Vert _{2}^{2}\Vert X_{1}^{\dag }\Vert _{2}^{2}}\left\| \sin \varTheta ( {\widetilde{X}}_{1},X_{1})\right\| _{F}^{2}\le \left\| {\widetilde{\varLambda }} _{2}^{-k}{\widetilde{Y}}_{2}^{*}X_{1}\varLambda _{1}^{1-l}-\widetilde{\varLambda }_{2}^{1-k}{\widetilde{Y}}_{2}^{*}X_{1}\varLambda _{1}^{-l}\right\| _{F}^{2}. \end{aligned}$$
(2.5)

It follows from Lemma 2.2 that there exists a permutation \(\tau \) of \( \langle r\rangle \) such that

$$\begin{aligned} \sigma _{\text {min}}^{2}\left( {\widetilde{Y}}_{1}^{*}X_{1}\right) \sum _{i=1}^{r}\left| \widetilde{ \lambda }_{i}^{-k}\lambda _{\tau (i)}^{1-l}-{\widetilde{\lambda }} _{i}^{1-k}\lambda _{\tau (i)}^{-l}\right| ^{2}\le \left\| {\widetilde{\varLambda }} _{1}^{-k}{\widetilde{Y}}_{1}^{*}X_{1}\varLambda _{1}^{1-l}-\widetilde{\varLambda }_{1}^{1-k}{\widetilde{Y}}_{1}^{*}X_{1}\varLambda _{1}^{-l}\right\| _{F}^{2}. \end{aligned}$$
(2.6)

By (2.3), (2.5) and (2.6) we obtain

$$\begin{aligned}&\frac{\left( \delta _{12}^{(l,k)}\right) ^{2}}{\Vert {\widetilde{Y}} _{2}^{\dag }\Vert _{2}^{2}\Vert X_{1}^{\dag }\Vert _{2}^{2}}\left\| \sin \varTheta ({\widetilde{X}}_{1},X_{1})\right\| _{F}^{2}+\sigma _{\text {min}}^{2}\left( \widetilde{Y }_{1}^{*}X_{1}\right) \sum _{i=1}^{r}\left| {\widetilde{\lambda }}_{i}^{-k}\lambda _{\tau (i)}^{1-l}-{\widetilde{\lambda }}_{i}^{1-k}\lambda _{\tau (i)}^{-l}\right| ^{2}\\&\quad \le \left\| {\widetilde{X}}^{-1}{\widetilde{A}}^{-k}\varDelta AA^{-l}X_{1}\right\| _{F}^{2}, \end{aligned}$$

which proves the desired bound. \(\square \)

Remark 2.1

Some existing bounds can be obtained from (2.1):

  • If we take \(X_{1}=X\) and \({\widetilde{X}}_{1}={\widetilde{X}}\), then \( \Vert \sin \varTheta ({\widetilde{X}}_{1},X_{1})\Vert _{F}=0\). For this case, (2.1) reduces to

    $$\begin{aligned} \sum _{i=1}^{n}\left| {\widetilde{\lambda }}_{i}^{-k}\lambda _{\tau (i)}^{1-l}-{\widetilde{\lambda }}_{i}^{1-k}\lambda _{\tau (i)}^{-l}\right| ^{2}\le \frac{\Vert {\widetilde{X}}^{-1}{\widetilde{A}}^{-k}\varDelta AA^{-l}X\Vert _{F}^{2}}{\sigma _{min}^{2}({\widetilde{X}}^{-1} X)}. \end{aligned}$$

    Notice that

    $$\begin{aligned} \sigma _{\text {min}}({\widetilde{X}}^{-1}X)=\Vert ({\widetilde{X}}^{-1}X)^{-1}\Vert _{2}^{-1}\ge \Vert {\widetilde{X}}\Vert _{2}^{-1}\Vert X^{-1}\Vert _{2}^{-1} \end{aligned}$$

    and

    $$\begin{aligned} \Vert {\widetilde{X}}^{-1}{\widetilde{A}}^{-k}\varDelta AA^{-l}X\Vert _{F}\le \Vert {\widetilde{X}}^{-1}\Vert _{2}\Vert X\Vert _{2} \Vert {\widetilde{A}}^{-k}\varDelta AA^{-l}\Vert _{F}. \end{aligned}$$

    Immediately, (1.7) follows from (2.1), which implies the proposed bound (2.1) is sharper than the bound (1.7).

  • By (2.2) and (2.5) it is easy to get the relative perturbation bound for eigenspaces:

    $$\begin{aligned} \Vert \sin \varTheta ({\widetilde{X}}_{1},X_{1})\Vert _{F}\le & {} \Vert {\widetilde{Y}}_{2}^{\dag }\Vert _{2}\Vert X_{1}^{\dag }\Vert _{2}\frac{\Vert {\widetilde{Y}}_{2}^{*}{\widetilde{A}}^{-k}\varDelta AA^{-l}X_{1}\Vert _{F}}{ \delta _{12}^{(l,k)}} \\\le & {} \kappa ({\widetilde{Y}}_{2})\kappa (X_{1})\frac{\Vert {\widetilde{A}} ^{-k}\varDelta AA^{-l}\Vert _{F}}{\delta _{12}^{(l,k)}}, \end{aligned}$$

    which is the bound (1.8).

  • If A and \({\widetilde{A}}\) are Hermitian, then X and \({\widetilde{X}}\) are unitary, and \(\sigma _{\text {min}}^{2}({\widetilde{X}}_{1}^{*}X_{1})=1-\Vert \sin \varTheta ({\widetilde{X}}_{1},X_{1})\Vert _{2}^{2}.\) Hence the bound (2.1) can be simplified as follows:

    $$\begin{aligned}&\left( \delta _{12}^{(l,k)}\right) ^{2}\Vert \sin \varTheta ({\widetilde{X}} _{1},X_{1})\Vert _{F}^{2}+\left( 1-\Vert \sin \varTheta ({\widetilde{X}} _{1},X_{1})\Vert _{2}^{2}\right) \sum _{i=1}^{r}\left| {\widetilde{\lambda }} _{i}^{-k}\lambda _{\tau (i)}^{1-l}-{\widetilde{\lambda }}_{i}^{1-k}\lambda _{\tau (i)}^{-l}\right| ^{2} \nonumber \\&\quad \le \Vert {\widetilde{A}}^{-k}\varDelta AA^{-l}X_{1}\Vert _{F}^{2}, \end{aligned}$$
    (2.7)

    which is a generalization of (2.3) of [16]. Comparing the bound (2.16) in [16] with (2.7), it is difficult to say which is sharper. When \(l=k=0,\) this bound reduces to (2.3) of [16]. In particular, by (2.7) we have

    $$\begin{aligned} \Vert \sin \varTheta ({\widetilde{X}}_{1},X_{1})\Vert _{F}\le \frac{\Vert {\widetilde{A}}^{-k}\varDelta AA^{-l}X_{1}\Vert _{F}}{\delta _{12}^{(l,k)}}. \end{aligned}$$
    (2.8)

    When \(l=k=0,\) the bound (2.8) is the classical \(\sin \varTheta \) theorem [3]. When \(l=k=\frac{1}{2},\) the bound (2.8) is givens as follows:

    $$\begin{aligned} \Vert \sin \varTheta ({\widetilde{X}}_{1},X_{1})\Vert _{F}\le \frac{\Vert {\widetilde{A}}^{-\frac{1}{2}}\varDelta AA^{-\frac{1}{2}}X_{1}\Vert _{F}}{\delta _{12}^{(\frac{1}{2},\frac{1}{2})}}. \end{aligned}$$
    (2.9)

    In [4], the authors obtained

    $$\begin{aligned} \Vert \sin \varTheta ({\widetilde{X}}_{1},X_{1})\Vert _{F}\le \frac{\Vert {\widetilde{A}}^{-\frac{1}{2}}\varDelta AA^{-\frac{1}{2}}\Vert _{F}}{\delta _{12}^{(\frac{1}{2},\frac{1}{2})}}, \end{aligned}$$
    (2.10)

    which can be derived from (2.9). Clearly, the bound (2.9) is sharper than the existing one (2.10).

3 Relative bounds for multiplicative perturbation

In this section, we consider the combined multiplicative perturbation bounds for diagonalizable matrices, i.e., the perturbed matrix \({\widetilde{A}}=D_{1}^{*}AD_{2}\), where \(D_{1}\) and \( D_{2}\) are nonsingular and close to the identity matrix. Assume that A and its perturbed matrix \({\widetilde{A}}\) are two \(n\times n\) nonsingular diagonalizable matrices with the eigendecompositions (1.1)–(1.3).

The proof of the following lemma can be given similar to the proof of Lemma 2.2 in [13].

Lemma 3.1

Let \(\varLambda _1\) and \({\widetilde{\varLambda }}_2 \) satisfy \({\widetilde{\varLambda }}_2 ^{1-k}B\varLambda _1 ^{-l}-{\widetilde{\varLambda }}_2^{-k}B\varLambda _1 ^{1-l}=\widetilde{ \varLambda _2 }E+F\varLambda _1\). Then

$$\begin{aligned} \Vert B\Vert _{F}\le \sqrt{\Vert E\Vert _{F}^{2}+\Vert F\Vert _{F}^{2}} /\rho _{12} ^{(l,k)}, \end{aligned}$$

where \(\varLambda _1\), \({\widetilde{\varLambda }}_2 \) and \(\rho _{12} ^{(l,k)}\) are given by (1.2), (1.3) and (1.6), respectively.

Lemma 3.2

Let \(\varLambda _1\) and \({\widetilde{\varLambda }}_1 \) satisfy \({\widetilde{\varLambda }}_1^{1-k}B\varLambda _1 ^{-l}-\widetilde{\varLambda }_1^{-k}B\varLambda _1 ^{1-l}=\widetilde{ \varLambda _1 }E+F\varLambda _1\), where \(\varLambda _1\) and \({\widetilde{\varLambda }}_1 \) are given by (1.2) and (1.3), respectively. Then there is a permutation \(\tau \) of \(\langle r\rangle \) such that

$$\begin{aligned} \sigma _{{\min } }^{2}(B)\sum _{i=1}^{r}\frac{|\lambda _{i}-{\widetilde{\lambda }} _{\tau (i)}|^{2} }{|\lambda _{i}|^{2l}|{\widetilde{\lambda }}_{\tau (i)}|^{2k}(| \lambda _{i}|^{2}+|{\widetilde{\lambda }}_{\tau (i)}|^{2})} \le \Vert E\Vert _{F}^{2}+\Vert F\Vert _{F}^{2}. \end{aligned}$$

Proof

Similar to the proof of Lemma 2.2 in [13], we obtain easily

$$\begin{aligned} \left| \frac{\lambda _{i}^{-l}{\widetilde{\lambda }}_{j}^{1-k}-\lambda _{i}^{1-l} {\widetilde{\lambda }}_{j}^{-k}}{\sqrt{\smash [b]{| \lambda _{i}|^{2}+|{\widetilde{\lambda }}_j|^{2}}}}\cdot b_{ji}\right| ^{2}\le |e_{ji}|^{2}+|f_{ji}|^{2}, \end{aligned}$$
(3.1)

from which one can deduce that

$$\begin{aligned} \sum _{i,j=1}^{r}\frac{|\lambda _{i}-{\widetilde{\lambda }} _j|^{2} }{|\lambda _{i}|^{2l}|{\widetilde{\lambda }}_j|^{2k}(| \lambda _{i}|^{2}+|{\widetilde{\lambda }}_j|^{2})}|b_{ji}|^{2}\le \Vert E\Vert _{F}^{2}+\Vert F\Vert _{F}^{2}. \end{aligned}$$
(3.2)

On the other hand, from the result given in the last line of [5], it is known that there is a permutation \(\tau \) of \(\langle r\rangle \) such that

$$\begin{aligned} \sigma _{\text {min} }^{2}(B)\sum _{i=1}^{r}\frac{|\lambda _{i}-{\widetilde{\lambda }} _{\tau (i)}|^{2} }{|\lambda _{i}|^{2l}|{\widetilde{\lambda }}_{\tau (i)}|^{2k}(| \lambda _{i}|^{2}+|{\widetilde{\lambda }}_{\tau (i)}|^{2})}\le \sum _{i,j=1}^{r}\frac{|\lambda _{i}-{\widetilde{\lambda }} _j|^{2} }{|\lambda _{i}|^{2l}|{\widetilde{\lambda }}_j|^{2k}(| \lambda _{i}|^{2}+|{\widetilde{\lambda }}_j|^{2})}|b_{ji}|^{2}, \end{aligned}$$

which together with (3.2) gives the desired bound. \(\square \)

The following bound (3.3) is the combined form for eigenvalues and eigenspaces of diagonalizable matrices.

Theorem 3.3

Let A and its perturbed matrix \( {\widetilde{A}}=D_{1}^{*}AD_{2}\) be two \(n\times n\) nonsingular diagonalizable matrices with the eigendecompositions (1.1)–(1.3), where \(D_{i}\) is nonsingular, \(i=1,2\). Then there is a permutation \(\tau \) of \(\langle r\rangle \) such that

$$\begin{aligned}&\frac{{ (\rho _{12} ^{(l,k)})}^{2}}{\Vert {\widetilde{Y}}_{2}^{\dag }\Vert _{2}^{2}\Vert X_{1}^{\dag }\Vert _{2}^{2}}\Vert \sin \varTheta (X_{1},\overset{ \sim }{X}_{1})\Vert _{F}^{2}+\sigma _{\min }^{2}(\widetilde{Y_1}^{*}X_{1})\sum _{i=1}^{r}\frac{|\lambda _{i}-{\widetilde{\lambda }} _{\tau (i)}|^{2} }{|\lambda _{i}|^{2l}|{\widetilde{\lambda }}_{\tau (i)}|^{2k}(| \lambda _{i}|^{2}+|{\widetilde{\lambda }}_{\tau (i)}|^{2})} \nonumber \\&\quad \le \Vert {\widetilde{X}}^{-1}{\widetilde{A}}^{-k}(I-D_{2}^{-1})A^{-l}X_{1}\Vert _{F}^{2}+\Vert {\widetilde{X}}^{-1}{\widetilde{A}}^{-k}(D_{1}^{*}-I)A^{-l}X_{1}\Vert _{F}^{2}. \end{aligned}$$
(3.3)

Proof

From \({\widetilde{A}}=D_{1}^{*}AD_{2}\) one may get

$$\begin{aligned} {\widetilde{A}}^{1-k}A^{-l}-{\widetilde{A}}^{-k}A^{1-l}={\widetilde{A}}\widetilde{A}^{-k}(I-D_{2}^{-1})A^{-l}+{\widetilde{A}}^{-k}(D_{1}^{*}-I)A^{-l}A. \end{aligned}$$
(3.4)

Multiplying by \({\widetilde{X}}^{-1}\) and \(X_{1}\) from the left and right of (3.4) respectively gives

$$\begin{aligned}&~~~~{\widetilde{\varLambda }}^{1-k}{\widetilde{X}}^{-1}X_{1}\varLambda _{1}^{-l}- {\widetilde{\varLambda }}^{-k}{\widetilde{X}}^{-1}X_{1}\varLambda _{1}^{1-l}\nonumber \\&\quad ={\widetilde{\varLambda }}{\widetilde{X}}^{-1}{\widetilde{A}} ^{-k}(I-D_{2}^{-1})A^{-l}X_{1}+{\widetilde{X}}^{-1}{\widetilde{A}} ^{-k}(D_{1}^{*}-I)A^{-l}X_{1}\varLambda _{1}. \end{aligned}$$
(3.5)

Let \(Q={\widetilde{X}}^{-1}X_{1},\) \(Z={\widetilde{X}}^{-1}{\widetilde{A}} ^{-k}(I-D_{2}^{-1})A^{-l}X_{1}\) and \({\widetilde{Z}}={\widetilde{X}}^{-1} {\widetilde{A}}^{-k}(D_{1}^{*}-I)A^{-l}X_{1}.\) By (1.4),

$$\begin{aligned} Q=\left( \begin{array}{c} {\widetilde{Y}}_{1}^{*}X_{1} \\ {\widetilde{Y}}_{2}^{*}X_{1} \end{array} \right) . \end{aligned}$$

Let

$$\begin{aligned} Z=\left( \begin{array}{c} Z_{1} \\ Z_{2} \end{array} \right) \quad \hbox { and }\quad {\widetilde{Z}}=\left( \begin{array}{l} {\widetilde{Z}}_{1} \\ {\widetilde{Z}}_{2} \end{array} \right) \end{aligned}$$

have the same block structure as Q. Rewriting (3.5) in the block form yields

$$\begin{aligned} \left( \begin{array}{c} {\widetilde{\varLambda }}_{1}^{1-k}{\widetilde{Y}}_{1}^{*}X_{1}\varLambda _{1}^{-l}-{\widetilde{\varLambda }}_{1}^{-k}{\widetilde{Y}}_{1}^{*}X_{1}\varLambda _{1}^{1-l} \\ {\widetilde{\varLambda }}_{2}^{1-k}{\widetilde{Y}}_{2}^{*}X_{1}\varLambda _{1}^{-l}-{\widetilde{\varLambda }}_{2}^{-k}{\widetilde{Y}}_{2}^{*}X_{1}\varLambda _{1}^{1-l} \end{array} \right) =\left( \begin{array}{l} {\widetilde{\varLambda }}_{1}Z_{1}+{\widetilde{Z}}_{1}\varLambda _{1} \\ {\widetilde{\varLambda }}_{2}Z_{2}+{\widetilde{Z}}_{2}\varLambda _{1} \end{array} \right) . \end{aligned}$$

Or equivalently,

$$\begin{aligned} {\widetilde{\varLambda }}_{1}^{1-k}{\widetilde{Y}}_{1}^{*}X_{1}\varLambda _{1}^{-l}-{\widetilde{\varLambda }}_{1}^{-k}{\widetilde{Y}}_{1}^{*}X_{1}\varLambda _{1}^{1-l}={\widetilde{\varLambda }}_{1}Z_{1}+{\widetilde{Z}}_{1}\varLambda _{1} \end{aligned}$$
(3.6)

and

$$\begin{aligned} {\widetilde{\varLambda }}_{2}^{1-k}{\widetilde{Y}}_{2}^{*}X_{1}\varLambda _{1}^{-l}-{\widetilde{\varLambda }}_{2}^{-k}{\widetilde{Y}}_{2}^{*}X_{1}\varLambda _{1}^{1-l}={\widetilde{\varLambda }}_{2}Z_{2}+{\widetilde{Z}}_{2}\varLambda _{1}. \end{aligned}$$
(3.7)

Applying Lemma 3.1 to (3.7) yields

$$\begin{aligned} \left( \rho _{1,2} ^{(l,k)}\right) ^{2}\Vert {\widetilde{Y}}_{2}^{*}X_{1}\Vert _{F}^{2}\le \Vert Z_{2}\Vert _{F}^{2}+\Vert {\widetilde{Z}}_{2}\Vert _{F}^{2}. \end{aligned}$$
(3.8)

It follows from Lemma 2.1 that

$$\begin{aligned} \frac{1}{\Vert {\widetilde{Y}}_{2}^{\dag }\Vert _{2}\Vert X_{1}^{\dag }\Vert _{2}}\Vert \sin \varTheta (X_{1},{\widetilde{X}}_{1})\Vert _{F}\le \Vert {\widetilde{Y}}_{2}^{*}X_{1}\Vert _{F}\,. \end{aligned}$$
(3.9)

By (3.8) and (3.9) we get

$$\begin{aligned} \frac{\left( \rho _{1,2} ^{(l,k)}\right) ^{2}}{\Vert {\widetilde{Y}}_{2}^{\dag }\Vert _{2}^{2}\Vert X_{1}^{\dag }\Vert _{2}^{2}}\Vert \sin \varTheta (X_{1}, {\widetilde{X}}_{1})\Vert _{F}^{2}\le \Vert Z_{2}\Vert _{F}^{2}+\Vert {\widetilde{Z}}_{2}\Vert _{F}^{2}. \end{aligned}$$
(3.10)

Applying Lemma 3.2 to (3.6) gives that there is a permutation \(\tau \) of \(\langle r\rangle \) so that

$$\begin{aligned} \sigma _{\min }^{2}({\widetilde{Y}}_{1}^{*}X_{1})\sum _{i=1}^{r}\frac{|\lambda _{i}-{\widetilde{\lambda }} _{\tau (i)}|^{2} }{|\lambda _{i}|^{2l}|{\widetilde{\lambda }}_{\tau (i)}|^{2k}(| \lambda _{i}|^{2}+|{\widetilde{\lambda }}_{\tau (i)}|^{2})} \le \Vert Z_{1}\Vert _{F}^{2}+\Vert {\widetilde{Z}} _{1}\Vert _{F}^{2}, \end{aligned}$$

which together with (3.10) gives that

$$\begin{aligned}&\frac{\left( \rho ^{(l,k)}\right) ^{2}}{\Vert {\widetilde{Y}}_{2}^{\dag }\Vert _{2}^{2}\Vert X_{1}^{\dag }\Vert _{2}^{2}}\Vert \sin \varTheta (X_{1}, {\widetilde{X}}_{1})\Vert _{F}^{2}+\sigma _{\min }^{2}({\widetilde{Y}}_{1}^{*}X_{1})\sum _{i=1}^{r}\frac{|\lambda _{i}-{\widetilde{\lambda }} _{\tau (i)}|^{2} }{|\lambda _{i}|^{2l}|{\widetilde{\lambda }}_{\tau (i)}|^{2k}(| \lambda _{i}|^{2}+|{\widetilde{\lambda }}_{\tau (i)}|^{2})} \\&\quad \le \Vert Z_{2}\Vert _{F}^{2}+\Vert {\widetilde{Z}}_{2}\Vert _{F}^{2}+\Vert Z_{1}\Vert _{F}^{2}+\Vert {\widetilde{Z}}_{1}\Vert _{F}^{2}=\Vert Z\Vert _{F}^{2}+\Vert {\widetilde{Z}}\Vert _{F}^{2} \\&\quad =\Vert {\widetilde{X}}^{-1}{\widetilde{A}}^{-k}(I-D_{2}^{-1})A^{-l}X_{1}\Vert _{F}^{2}+\Vert {\widetilde{X}}^{-1}{\widetilde{A}}^{-k}(D_{1}^{*}-I)A^{-l}X_{1}\Vert _{F}^{2}. \end{aligned}$$

This proves the theorem. \(\square \)

Remark 3.1

From Theorem 3.3 and its proof, we can derive some existing bounds:

  • If we take \(X_{1}=X\) and \({\widetilde{X}}_{1}={\widetilde{X}}\), then \( \Vert \sin \varTheta (X_{1},{\widetilde{X}}_{1})\Vert _{2}=\Vert \sin \varTheta (X_{1},{\widetilde{X}}_{1})\Vert _{F}=0\) and the bound (3.3) implies that

    $$\begin{aligned}&\sqrt{\sum _{i=1}^{n} \frac{|\lambda _i-{\tilde{\lambda }}_{\tau (i)}|^2}{|\lambda _i|^{2l} |{\tilde{\lambda }}_{\tau (i)}|^{2k} (|\lambda _i|^2+|{\tilde{\lambda }}_{\tau (i)}|^2)}}\ \\&\quad \le \frac{1}{\sigma _{\min }({\widetilde{X}}^{-1}X)}\sqrt{||{\widetilde{X}} ^{-1}{\widetilde{A}}^{-k}(I-D_{2}^{-1})A^{-l}X\Vert _{F}^{2}+\Vert \widetilde{X }^{-1}{\widetilde{A}}^{-k}(D_{1}^{*}-I)A^{-l}X\Vert _{F}^{2}} \\&\quad \le \Vert {\widetilde{X}}X^{-1}\Vert _{2}\Vert X\Vert _{2}\Vert {\widetilde{X}} ^{-1}\Vert _{2}\sqrt{\Vert {\widetilde{A}}^{-k}(I-D_{2}^{-1})A^{-l}\Vert _{F}^{2}+\Vert {\widetilde{A}}^{-k}(D_{1}^{*}-I)A^{-l}\Vert _{F}^{2}} \\&\quad \le \kappa (X)\kappa ({\widetilde{X}})\sqrt{\Vert {\widetilde{A}} ^{-k}(I-D_{2}^{-1})A^{-l}\Vert _{F}^{2}+\Vert {\widetilde{A}}^{-k}(D_{1}^{*}-I)A^{-l}\Vert _{F}^{2}}. \end{aligned}$$

    When \(l=k=0,\) the above bound reduces to the one given by Li [10].

  • By (3.10) we may deduce the bound for eigenspaces. In fact,

    $$\begin{aligned}&\frac{\left( \rho _{1,2} ^{(l,k)}\right) ^{2}}{\Vert {\widetilde{Y}}_{2}^{\dag }\Vert _{2}^{2}\Vert X_{1}^{\dag }\Vert _{2}^{2}}\Vert \sin \varTheta (X_{1}, {\widetilde{X}}_{1})\Vert _{F}^{2}\\&\quad \le \Vert {\widetilde{Y}}_{2}^{*} {\widetilde{A}}^{-k}(I-D_{2}^{-1})A^{-l}X_{1}\Vert _{F}^{2}+\Vert {\widetilde{Y}} _{2}^{*}{\widetilde{A}}^{-k}(D_{1}^{*}-I)A^{-l}X_{1}\Vert _{F}^{2}, \end{aligned}$$

    and thus

    $$\begin{aligned}&\Vert \sin \varTheta (X_{1},{\widetilde{X}}_{1})\Vert _{F}\nonumber \\&\quad \le \frac{\kappa (X_{1})\kappa ({\widetilde{Y}}_{2})\sqrt{\Vert {\widetilde{A}} ^{-k}(I-D_{2}^{-1})A^{-l}\Vert _{F}^{2}+\Vert {\widetilde{A}}^{-k}(D_{1}^{*}-I)A^{-l}\Vert _{F}^{2}}}{\rho _{12} ^{(l,k)}}. \qquad \qquad \end{aligned}$$
    (3.11)

    When \(l=k=0,\) the above bound is identical to the bound given in Remark 3.3 of [13].

  • If A and \({\widetilde{A}}=D^{*}AD\) are Hermitian, then X and \({\widetilde{X}}\) are unitary, and \(\sigma _{\text {min}}^{2}({\widetilde{X}}_{1}^{*}X_{1})=1-\Vert \sin \varTheta ({\widetilde{X}}_{1},X_{1})\Vert _{2}^{2}.\) Hence the bound (3.3) can be rewritten as the following form:

    $$\begin{aligned}&\left( \rho _{1,2} ^{(l,k)}\right) ^{2}\Vert \sin \varTheta ({\widetilde{X}}_{1},X_{1})\Vert _{F}^{2}+(1-\Vert \sin \varTheta ({\widetilde{X}}_{1},X_{1})\Vert _{2}^{2})\varepsilon (\lambda _{i}, {\widetilde{\lambda }} _{\tau (i)}) \nonumber \\&\quad \le \Vert {\widetilde{A}}^{-k}(I-D^{-1})A^{-l}X_{1}\Vert _{F}^{2}+\Vert {\widetilde{A}}^{-k}(D^{*}-I)A^{-l}X_{1}\Vert _{F}^{2}, \end{aligned}$$
    (3.12)

    where by \(\varepsilon (\lambda _{i}, {\widetilde{\lambda }} _{\tau (i)})\) we denote the sum

    $$\begin{aligned} \sum _{i=1}^{r}\frac{|\lambda _{i}-{\widetilde{\lambda }} _{\tau (i)}|^{2} }{|\lambda _{i}|^{2l}|{\widetilde{\lambda }}_{\tau (i)}|^{2k}(| \lambda _{i}|^{2}+|{\widetilde{\lambda }}_{\tau (i)}|^{2})}. \end{aligned}$$

    If \(l=k=0,\) then the bound (3.12) reduces to (2.2) of [14].

4 Numerical experiments

From Sects. 2 and 3, the perturbation bounds (2.1) and (3.3) not only generalize some existing perturbation bounds but also can be used to produce new perturbation bounds. In this section, we will take some matrices from the University of Florida Sparse Matrix collection to test the new perturbation bounds (2.8) and (3.11), respectively. The test matrices are described in Table  1, where we denote Symmetric Positive Definite and Hermite Positive Definite by SPD and HPD, respectively. Note that these test matrices have the large 2-norm condition numbers from the magnitude order \(10^5\) to \(10^{12}\).

Table 1 Name, sources and properties of the \(n\times n\) test matrices A
Full size table

For all test matrices, the perturbations \(\varDelta A\) in (2.8) and \(D_i (i=1,2)\) in (3.11) are generated randomly by the MATLAB commands sprandsym and sprand, which are explained in detail as follows:

  • sprandsym(A): giving a sparse symmetric random matrix with the same structure as A, whose nonzero entries lie in the interval (0, 1);

  • sprand(m, n, d): giving an \(m\times n\) sparse random matrix with approximately dmn nonzero entries in the interval (0, 1) each time;

  • sprand(A): giving a sparse random matrix with the same structure as A each time, whose nonzero entries locate in the interval (0, 1).

We run the above commands and then generate the additive and multiplicative perturbations with small enough \(\varepsilon \), which are given as follows:

  • For the bound (2.8), let \(\varDelta A=\varepsilon B\) with \(B=sprandsym(A)\);

  • For the bound (3.11), let \(D_i=I+\varepsilon E_i\) (\(i=1,2\)). We distinguish the following cases:

    • For the test matrices from bcsstk14, mhd1280b and nasa1824, let

      $$\begin{aligned} \alpha =sprand(n, 1, 1e-2) \quad \hbox { and} \quad E_i=\mathrm {diag}(\alpha ) \end{aligned}$$

      where \(\mathrm {diag}(\alpha )\) is a diagonal matrix with the elements of the vector \(\alpha \) on the main diagonal;

    • For the test matrix from plskz362, let \(E_i=sprand(A)\). For this case, \(D_i\) is not necessarily diagonal.

As pointed out in Sects. 2–3, the derived bounds (2.8) and (3.11) are also generalizations of some existing bounds. In Tables 2 and 3 and Fig. 1, we give some comparison results of the bounds (2.8) and (3.11) with different values of l and k. For simplicity, we use notations \(\text {Pb}_{\text {new}}(l,k)\) and \(\text {Pb}_{\text {old}}(l,k)\) to denote a new bound and an existing one, respectively, which can be derived by the proposed bounds (2.8) or (3.11). In other words, both \(\text {Pb}_{\text {new}}(l,k)\) and \(\text {Pb}_{\text {old}}(l,k)\) are exactly (2.8) or (3.11) with specific values for l and k. In addition, the notation \(\text {Pb}_{\text {old}}\) is used to denote the existing bound (2.10). Here we choose the quasi-optimal parameters \(l_*\) and \(k_*\) in the bounds (2.8) and (3.11) by the following methods:

  • For the additive perturbation bound (2.8), \(l_*\) and \(k_*\) are chosen by taking \((l, k)=(0, 0)\), (1, 0), (0, 1) and (0.5, 0.5), respectively, so that the associated perturbation bound attains the minimum.

  • For the multiplicative perturbation bound (3.11), \(l_{*}\) and \(k_{*}\) are obtained experimentally by minimizing the bound (3.11) in the interval [0, 0.5].

Table 2 The additive perturbation bound (2.8) for SpaceStation9: \(\varepsilon =10^{-5}\), \((l_*, k_*)=(1, 0)\)
Full size table
Table 3 The multiplicative perturbation bound (3.11) for the matrices A: \(r=300\), \(\varepsilon =10^{-5}\)
Full size table
Fig. 1
figure 1

Additive perturbation bound (2.8) for the test matrix SpaceStation with \(r=500\): \(l_*=1\), \(k_*=0\)

Full size image

By Table  2 we report the numerical results of the bound (2.8) for the test matrices from the optimal control field with the different sizes \(X_1\) and \(X_2\), which are shown by the different r, in which one may see that the perturbation bound (2.8) with the quasi-optimal parameters \(l_*\) and \(k_*\) is always tighter than the associated existing ones \(\text {Pb}_{\text {old}}(0,0)\) and \(\text {Pb}_{\text {old}}\). The latter two ones are exactly the \(\sin \varTheta \) theorem in [3] and the bound in [4], respectively.

In Fig. 1 we denote by NormS the real value of \(\Vert \sin \varTheta ({\tilde{X}}_1,X_1)\Vert _F\). From Fig. 1 it is known that the new bound \(\text {Pb}_{\text {new}}(l_*,k_*)\) is still sharper than the existing ones for the different values of the perturbation \(\varepsilon \). Moreover, Fig. 1 implies that the bound (2.8) with \(l=k=0.5\) always outperforms the associated existing one (2.10), which further confirms the theoretical analysis in Sect. 2.

In order to verify the effectiveness of the multiplicative perturbation bound (3.11), we test the different matrices with \(r=300\) and \(\varepsilon =10^{-5}\), where the bound \(\text {Pb}_{\text {old}}(0,0)\) is given by Remark 3.3 of [13]. As expected, the quasi-optimal bound \(\text {Pb}_{\text {new}}(l_{*}, k_{*})\) from (3.11) is always sharper than the existing bound \(\text {Pb}_{\text {old}}(0,0)\).

Remark 4.1

From numerical results given in Tables 2 and 3 and Fig. 1, we can always obtain the sharper new bounds by taking l and k. In particular, we test the additive bound (2.8) and the multiplicative bound (3.11) by a large number of examples, which show that quasi-optimal parameters are given by \((l_*,k_*)=(1, 0)\) or (0, 1) for (2.8) and \(|l_*-k_*|\le 0.1\) for (3.11) when \(l, k\in [0, 0.5]\). However, we are not able to present some theoretical results for the optimal parameters in the proposed perturbation bounds. This remains open.

5 Conclusions

In this paper, we have proposed two relative perturbation bounds (2.1) and (3.3), which provide a general framework of relative bounds for additive and multiplicative perturbations for eigenpairs of diagonalizable matrices, respectively. With suitable choices of eigenspaces or the parameters l and k, the new bounds (2.1) and (3.3) not only cover some classical perturbation bounds as their special cases, but also yield some new perturbation bounds, for example, the new bounds (2.8) and (3.11). We have shown that the bounds (2.1) and (3.3) may improve some existing ones. The numerical experiments reveal that new perturbation bounds are sharper than the existing ones. In the future we will consider to get the optimal parameters l and k in the bounds (2.1) and (3.3) for some special structure matrices.