Is the Similarity Matrix S of A always the Identity Matrix?

[td21]

for a matrix similar to itself:
$A=SAS^{-1}$

A is nonzero and not identity.

So is the matrix S must be identity matrix? Can it be non-identity matrix? Thank you.

 

[tiny-tim]

hi td21!

(you need to use {} if there's more than one character after ^ or _ )

what if A is diagonal?

 

[td21]

hi td21!

(you need to use {} if there's more than one character after ^ or _ )

what if A is diagonal?

Thank you very much for answering! Yes, if A is diagonal S can be non-identity!
But what if A is not diagonal? Can S be non-identity?(I believe so, but i cannot give a proof. Is it possible to give a proof?)

Also i believe that in any cases(A being diagonal or not), S has to be diagonal. Is this true?

 

[micromass]

Hi td21!

What if

$$A=\left(\begin{array}{cc} a & b\\ 0 & a\\ \end{array}\right)$$

and

$$S=\left(\begin{array}{cc} 1 & 2\\ 0 & 1\\ \end{array}\right)$$

 

[blue_raver22]

I would say that the matrix S does not necessarily have to be the identity matrix. It can be any non-singular matrix that satisfies the equation A=SAS^{-1}. This means that there are infinitely many possible matrices S that can make A similar to itself. However, if A is nonzero and not the identity matrix, it is likely that S will also be a non-identity matrix. It is important to note that the concept of similarity in matrices is not unique, and there can be multiple matrices that are similar to a given matrix A.