### On Similarity of an Arbitrary Matrix to a Block Diagonal Matrix

#### Abstract

Let an $n\by n$ -matrix $A$ have

$m< n$ $(m\ge 2)$ different eigenvalues

$\la_j$ of the algebraic multiplicity $\mu_j$ $(j=1, ..., m)$.

It is proved that there are $\mu_j\by \mu_j$-matrices $A_j,$ each of which has a unique eigenvalue $\la_j$, such

that $A$ is similar to the block-diagonal matrix $\hat D=\diag\;(A_1, A_2, ..., A_m)$. I.e. there

is an invertible matrix $T$, such that

$T\mi A T=\hat D$. Besides, a sharp bound for the

number $\ka_T:=\|T\|\|T\mi\|$ is derived.

As applications of these results we obtain norm estimates for matrix functions

non-regular on the convex hull of the spectra. These estimates generalize and refine the previously published results. In addition,

a new bound for the spectral variation of matrices is derived.

In the appropriate situations it

refines the well known bounds.

$m< n$ $(m\ge 2)$ different eigenvalues

$\la_j$ of the algebraic multiplicity $\mu_j$ $(j=1, ..., m)$.

It is proved that there are $\mu_j\by \mu_j$-matrices $A_j,$ each of which has a unique eigenvalue $\la_j$, such

that $A$ is similar to the block-diagonal matrix $\hat D=\diag\;(A_1, A_2, ..., A_m)$. I.e. there

is an invertible matrix $T$, such that

$T\mi A T=\hat D$. Besides, a sharp bound for the

number $\ka_T:=\|T\|\|T\mi\|$ is derived.

As applications of these results we obtain norm estimates for matrix functions

non-regular on the convex hull of the spectra. These estimates generalize and refine the previously published results. In addition,

a new bound for the spectral variation of matrices is derived.

In the appropriate situations it

refines the well known bounds.

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