Chapter 6: Bounds for Condition Numbers of Diagonalizable Matrices

Abstract:

An eigenvalue is said to be simple, if its geometric multiplicity is equal to one. In this chapter 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

where $\widehat{D}$ is a normal matrix. Besides, A is called a diagonalizable matrix. The condition number $k(A,T):=\Vert T\Vert \Vert T^{-1}\Vert$ is very important for various applications. We obtain a bound for the condition number and discuss applications of that bound to matrix functions and equations with diagonalizable matrices.