Can all invertible matrices be diagonalized?

[vvgobre]

For a matrix [X] ,

Is there anyway to calculate the Error function of matrix or Erf[X] ?

Any possible solution to above will highly appreciated! :)

 

[Stephen Tashi]

The general way to use the definition of a single variable function f(x) to define a matrix function f(X) is to take a power series for f(x) (like the McLaurin series) and substitute the matrix X for the variable x in it. Of course, this only makes sense if the power series in the matrix converges to some matrix.

 

[vvgobre]

Thank you Stephen. :)

Recently i learn that if you have a matrix [A]

by diagonalization as A= V D V^-1 :D is diagonal matrix , V is eigenvector
can be use to calculate nth power (A^n) of matrix A as

A^(1/2) = V . D^(1/2) . V-1

source:http://en.wikipedia.org/wiki/Square_root_of_a_matrix

So basically i am using FORTRAN and i need to write three routine for
Erf[matrix], exp[matrix], [matrix]^a where a is real.

So Is it mathematically "true" ? if i generalize aforementioned algorithm to my three "needs" as

1) Erf[A] = V . Erf[D] . V-1

2) exp[A] = V . exp[D] . V-1

3) [A]^a = V . [D]^a . V-1 ; a is real number

?

 

[Stephen Tashi]

Yes, those are correct. Of course, you can't diagonalize all matrices. Were you writing your algorithm to apply only to those that can be diagonalized?

By the way, a modern book on this subject is "Functions Of Matrices" by Nicholas Higham published by SIAM.

 

[vvgobre]

Thank you Stephen for prompt reply and reference too.

Yes i do check whether the matrix is invertible or not.

 

[Stephen Tashi]

Some invertible matrices are not diagonalizable. And some diagonalizable matrices are not invertible. (Zero is a legitimate eigenvalue.)