# Similarity and Digonalization in matrices

• Jan 22nd 2008, 09:19 PM
somestudent2
Similarity and Digonalization in matrices
a) If A and B are invertible matrices show that AB and BA are similar

b) Let A be an invertible matrix. Prove that if A is digonalizable, so is its inverse.(A^-1)

Thank You
• Jan 23rd 2008, 02:48 AM
Opalg
Quote:

Originally Posted by somestudent2
a) If A and B are invertible matrices show that AB and BA are similar

\$\displaystyle BA = A^{-1}(AB)A\$.

Quote:

Originally Posted by somestudent2
b) Let A be an invertible matrix. Prove that if A is digonalizable, so is its inverse.(A^-1)

If A is diagonalisable then \$\displaystyle P^{-1}AP = D\$. What happens when you take the inverse of both sides of that equation?
• Jan 23rd 2008, 07:45 AM
somestudent2
Hi

for the second one by taking inverse on both sides we arive at

P^-1A^-1P = D^-1 ?

And this proves it (does it matter that Diagonal matrix is inversed now)? Or I need to make some additional calculations?

Thanks
• Jan 23rd 2008, 10:10 AM
Opalg
Quote:

Originally Posted by somestudent2
for the second one by taking inverse on both sides we arive at

P^-1A^-1P = D^-1 ?

And this proves it (does it matter that Diagonal matrix is inversed now)? Or I need to make some additional calculations?

If a diagonal matrix has an inverse then the inverse is also diagonal. So all you need to do is to explain why D is invertible. (Reason: the diagonal elements of D are the eigenvalues of A and, since A is invertible, none of them can be zero.)