Pseudospectra localizations for generalized tensor eigenvalues to seek more positive definite tensors

Abstract

In this paper, we present the pseudospectrum for generalized tensor eigenvalues, and a set to locate this pseudospectrum. By the relations between H-eigenvalues (Z-eigenvalues) of tensors and generalized tensor eigenvalues, a pseudospectral localization for H-eigenvalues (Z-eigenvalues, respectively) is given to seek positive definite tensors surrounding a positive definite tensor.

Appendix

We shall show the relations of US-eigenvalues and generalized tensor eigenvalues. Let ${\mathcal {A}} \in {\mathbb {C}}^{[m,n]}$ be a complex symmetric tensor, and let $\widehat{{\mathcal {A}}} \in {\mathbb {C}}^{[m,2n]}$ with

$$\begin{aligned} \widehat{{\mathcal {A}} }\left( 1:n,\ldots , 1:n\right) ={\mathcal {A}}^*,~ \widehat{{\mathcal {A}} }\left( n+1:2n,\ldots , n+1:2n\right) ={\mathcal {A}}, \end{aligned}$$

and $\widehat{\mathbf{x }}= \left( \mathbf{x }^\top ,(\mathbf{x }^*)^\top \right) ^\top =({\widehat{x}}_1,\ldots , {\widehat{x}}_{2n})^\top$. When m is even, let

$$\begin{aligned} {\mathcal {J}}=({\mathcal {J}}_{i_1\cdots i_m}) \in {\mathbb {R}}^{[m,2n]} \end{aligned}$$

(13)

be a tensor such that

$$\begin{aligned} ({\mathcal {J}} \mathbf{y }^{m-1})_i= \left\{ \begin{array}{c@{\quad }l} y_{i+n}(y_1y_{n+1}+y_2y_{n+2}+\cdots y_ny_{2n})^{\frac{m-2}{2}}, \quad &{}i=1,2,\ldots ,n, \\ y_{i-n}(y_1y_{n+1}+y_2y_{n+2}+\cdots y_ny_{2n})^{\frac{m-2}{2}}, \quad &{}i=n+1,n+2,\ldots ,2n, \end{array} \right. \end{aligned}$$

for an arbitrary vector $\mathbf{y }\in {\mathbb {C}}^{2n}$. Since

$$\begin{aligned} {\widehat{x}}_1{\widehat{x}}_{n+1}+\cdots +{\widehat{x}}_n{\widehat{x}}_{2n}= x_1x_{1}^*+\cdots +x_nx_{n}^*=||\mathbf{x }||^2_2=1, \end{aligned}$$

we rewrite the definition of US-eigenvalues into

$$\begin{aligned} \widehat{{\mathcal {A}}} \widehat{\mathbf{x }}^{m-1}= \lambda {\mathcal {J}} \widehat{\mathbf{x }}^{m-1}. \end{aligned}$$

(14)

This implies that if $\lambda$ is an US-eigenvalue of an even complex symmetric tensor ${\mathcal {A}}$, then $\lambda \in \sigma ( \widehat{{\mathcal {A}}}, {\mathcal {J}})$, also see Page 1074-1075 of Ding and Wei (2015).

The $\epsilon$-pseudo-spectrum for US-eigenvalues of a complex symmetric tensor ${\mathcal {A}}\in {\mathbb {C}}^{[m,n]}$ is defined as follows:

$$\begin{aligned} \varLambda _\epsilon ^{US}({\mathcal {A}},|| \cdot ||):= & {} \left\{ \lambda \in {\mathbb {C}} : ({\mathcal {A}}+{\mathcal {E}})^*\mathbf{x }^{m-1}=\lambda \mathbf{x }^{*}, ~({\mathcal {A}}+ {\mathcal {E}})\mathbf{x }^{*m-1} =\lambda \mathbf{x },\right. \\&\left. ||\mathbf{x }||_2=1~{\mathrm{for~ some ~symmetric ~tensor }} ~{\mathcal {E}}\in {\mathbb {C}}^{[m,n]} \right. \\&\left. {\mathrm{with}}~ ||{\mathcal {E}}||\le \epsilon , \mathbf{x } \in {\mathbb {C}}^{n}\right\} . \end{aligned}$$

Clearly, $\varLambda _0^{US}({\mathcal {A}},|| \cdot ||)$ can be denoted the spectrum for US-eigenvalues of a complex symmetric tensor ${\mathcal {A}}$.

Based the relations between US-eigenvalues with generalized tensor eigenvalues, we give the following set to locate $\epsilon$-pseudo-spectrum for US-eigenvalues relative to the tensor infinity norm: $\varLambda _\epsilon ^{US}({\mathcal {A}},|| \cdot ||_\infty )$.

Theorem 3

Let ${\mathcal {A}}=(a_{i_1\cdots i_m}) \in {\mathbb {C}}^{[m,n]}$ be symmetric with an even m, and $\epsilon \ge 0$. Then

$$\begin{aligned} \varLambda _\epsilon ^{US}({\mathcal {A}},|| \cdot ||_\infty ) \subseteq \varGamma ^{US}({\mathcal {A}},\epsilon ){:=}\bigcup \limits _{i\in N} \varGamma _i^{US}({\mathcal {A}},\epsilon ) , \end{aligned}$$

where

$$\begin{aligned} \varGamma _i^{US}({\mathcal {A}},\epsilon ){:=} \left\{ z\in {\mathbb {C}} : | a_{i\cdots i}|-r_{i}({\mathcal {A}})\le |\lambda | n^{\frac{m-2}{2}}+ \epsilon \right\} . \end{aligned}$$

Proof

Let $\lambda \in \varLambda _\epsilon ^{US}({\mathcal {A}},|| \cdot ||_\infty )$. Then there is a symmetric tensor ${\mathcal {E}}=({\mathcal {E}}_{i_1\cdots i_m})\in {\mathbb {C}}^{[m,n]}$ and a nonzero vector $\mathbf{x }=(x_1,\ldots ,x_n)^\top \in {\mathbb {C}}^{n}$ such that

$$\begin{aligned} ({\mathcal {A}}+{\mathcal {E}})^*\mathbf{x }^{m-1}=\lambda \mathbf{x }^{*}, ({\mathcal {A}}+ {\mathcal {E}})\mathbf{x }^{*m-1} =\lambda \mathbf{x }, ||\mathbf{x }||_2=1. \end{aligned}$$

By (14), we have

$$\begin{aligned} \lambda \in \sigma \left( \widehat{{\mathcal {A}}+{\mathcal {E}}}, {\mathcal {J}}\right) , \end{aligned}$$

where $\widehat{{\mathcal {A}}+{\mathcal {E}} } =({\widehat{a}}_{i_1\cdots i_m})\in {\mathbb {C}}^{[m,2n]}$ with

$$\begin{aligned} \widehat{{\mathcal {A}}+{\mathcal {E}} }\left( 1:n,\ldots , 1:n\right) =({\mathcal {A}}+{\mathcal {E}})^*,~ \widehat{{\mathcal {A}}+{\mathcal {E}} }\left( n+1:2n,\ldots , n+1:2n\right) ={\mathcal {A}}+{\mathcal {E}}, \end{aligned}$$

and ${\mathcal {J}}$ is defined as in (13). Furthermore, by Corollary 1 it follows that

$$\begin{aligned} \lambda \in \varGamma \left( \widehat{{\mathcal {A}}+{\mathcal {E}}}, {\mathcal {J}}\right) =\bigcup \limits _{i\in \{1,\ldots ,2n\}} \varGamma _i\left( \widehat{{\mathcal {A}}+{\mathcal {E}}}, {\mathcal {J}}\right) . \end{aligned}$$

Then there is an index $i_0\in \{1,2,\ldots , 2n\}$ such that

$$\begin{aligned} | \lambda {\mathcal {J}}_{i_0\cdots i_0}- {\widehat{a}}_{i_0\cdots i_0}| \le \sum \limits ^{2n}_{\begin{array}{c} i_2,\ldots ,i_m=1, \\ (i_2,\ldots ,i_m)\ne (i_0,\ldots ,i_0) \end{array}} |\lambda {\mathcal {J}}_{i_0i_2\cdots i_m} - {\widehat{a}}_{i_0i_2\cdots i_m}|. \end{aligned}$$

(15)

For the tensor ${\mathcal {J}}$, it is not difficult to see that

$$\begin{aligned} {\mathcal {J}}_{i\cdots i}=0, \quad i=1,2\ldots ,2n, \end{aligned}$$

(16)

and that for any $i_k\in \{1,2,\ldots ,n\}$ and any $j_k \in \{n+1,n+2,\ldots ,2n\}, k=1,\ldots ,m$,

$$\begin{aligned} {\mathcal {J}}_{i_1i_2\cdots i_m}={\mathcal {J}}_{j_1j_2\cdots j_m}=0. \end{aligned}$$

(17)

On the other hand, for the tensor $\widehat{{\mathcal {A}}+{\mathcal {E}} } =({\widehat{a}}_{i_1\cdots i_m})$, it is easy to see that for each $i\in \{1,2,\ldots ,n\}$,

$$\begin{aligned} {\widehat{a}}_{ii_2\cdots i_m}=0, ~ \mathrm{for} ~\mathrm{some}~i_k\in \{n+1,\ldots ,2n\}, k=2,\ldots ,m, \end{aligned}$$

(18)

and that for each $i\in \{n+1,n+2,\ldots ,2n\}$,

$$\begin{aligned} {\widehat{a}}_{ii_2\cdots i_m}=0, ~ \mathrm{for} ~\mathrm{some}~i_k\in \{1,\ldots ,n\}, k=2,\ldots ,m. \end{aligned}$$

(19)

Combining (15), (16), (17), (18), (19) and Remark 2, we have that if $i_0\in \{1,2,\ldots ,n\}$, then

$$\begin{aligned} | {\widehat{a}}_{i_0\cdots i_0}|\le \sum \limits ^n_{\begin{array}{c} i_2,\ldots ,i_m=1, \\ (i_2,\ldots ,i_m)\ne (i_0,\ldots ,i_0) \end{array}} | {\widehat{a}}_{i_0i_2\cdots i_m}| + \sum \limits _{(i_2,\ldots ,i_m)\in \varDelta _{i_0}({\mathcal {J}})} |\lambda {\mathcal {J}}_{i_0i_2\cdots i_m}|, \end{aligned}$$

(20)

and that if $i_0\in \{n+1,n+2,\ldots ,2n\}$, then

$$\begin{aligned} | {\widehat{a}}_{i_0\cdots i_0}|\le \sum \limits ^{2n}_{\begin{array}{c} i_2,\ldots ,i_m=n+1, \\ (i_2,\ldots ,i_m)\ne (i_0,\ldots ,i_0) \end{array}} | {\widehat{a}}_{i_0i_2\cdots i_m}| + \sum \limits _{(i_2,\ldots ,i_m)\in \varDelta _{i_0}({\mathcal {J}})} |\lambda {\mathcal {J}}_{i_0i_2\cdots i_m}|. \end{aligned}$$

(21)

We now consider the equivalent form of

$$\begin{aligned} \sum \limits _{(i_2,\ldots ,i_m)\in \varDelta _{i_0}({\mathcal {J}})} |\lambda {\mathcal {J}}_{i_0i_2\cdots i_m}|. \end{aligned}$$

Taking $\mathbf{y }=\mathbf{e }=(1,1,\ldots ,1)^\top \in {\mathbb {R}}^{2n}$, then by (13) we have

$$\begin{aligned} ({\mathcal {J}} \mathbf{e }^{m-1})_{i}= \sum \limits _{i_2,\ldots ,i_m\in N} {\mathcal {J}}_{ii_2\cdots i_m}= \sum \limits _{(i_2,\ldots ,i_m)\in \varDelta _{i}({\mathcal {J}})} {\mathcal {J}}_{ii_2\cdots i_m}= n^{\frac{m-2}{2}}, \end{aligned}$$

whenever $i=1,2\ldots ,2n$. Hence,

$$\begin{aligned} \sum \limits _{(i_2,\ldots ,i_m)\in \varDelta _{i_0}({\mathcal {J}})} |\lambda {\mathcal {J}}_{i_0i_2\cdots i_m}|=|\lambda | n^{\frac{m-2}{2}}. \end{aligned}$$

(22)

Furthermore, note that

$$\begin{aligned} {\widehat{a}}_{i_1i_2\cdots i_m}= \left( a_{i_1i_2\cdots i_m}+ {\mathcal {E}}_{i_1i_2\cdots i_m}\right) ^*,~ \mathrm{for} ~ i_1,\ldots ,i_m\in \{1,2,\ldots ,n\} \end{aligned}$$

and

$$\begin{aligned} {\widehat{a}}_{i_1i_2\cdots i_m}=a_{i_1i_2\cdots i_m}+ {\mathcal {E}}_{i_1i_2\cdots i_m}, ~\mathrm{for}~ i_1,\ldots ,i_m\in \{n+1,n+2,\ldots ,2n\}. \end{aligned}$$

Then (20), (21), and (22) together give

$$\begin{aligned} | a_{j_0\cdots j_0}+{\mathcal {E}}_{j_0\cdots j_0}|\le \sum \limits _{\begin{array}{c} i_2,\ldots ,i_m\in N, \\ (i_2,\ldots ,i_m)\ne (j_0,\ldots ,j_0) \end{array}} | a_{j_0i_2\cdots i_m}+ {\mathcal {E}}_{j_0i_2\cdots i_m}| + |\lambda | n^{\frac{m-2}{2}}, \end{aligned}$$

where

$$\begin{aligned} j_0=\left\{ \begin{array}{ll} i_0, &{} \quad \mathrm{if} ~i_0\in \{1,2,\ldots , n\},\\ i_0-n,&{} \quad \mathrm{if} ~i_0\in \{n+1,n+2,\ldots , 2n\}. \end{array}\right. \end{aligned}$$

(23)

Using the triangle inequality yields

$$\begin{aligned} | a_{j_0\cdots j_0}|-r_{j_0}({\mathcal {A}})\le |\lambda | n^{\frac{m-2}{2}}+ R_{j_0}({\mathcal {E}}). \end{aligned}$$

Since $R_{i_0}({\mathcal {E}})\le || {\mathcal {E}}||_\infty \le \epsilon$, we have

$$\begin{aligned} | a_{j_0\cdots j_0}|-r_{j_0}({\mathcal {A}})\le |\lambda | n^{\frac{m-2}{2}}+ \epsilon , \end{aligned}$$

and thus $\lambda \in \varGamma ^{US}_{j_0}({\mathcal {A}},\epsilon )\in \varGamma ^{US}({\mathcal {A}},\epsilon )$. The conclusion follows. $\square$

Taking $\epsilon =0$ in Theorem 3 yields Theorem 2.