# Are Similar Matrices Necessarily Congruent?

• Jan 17th 2008, 01:02 AM
kevin_chn
Are Similar Matrices Necessarily Congruent?
As two similar matrices have a same set of eigenvalues, are they necessarily congruent?

Thank you~
• Jan 17th 2008, 01:24 AM
kevin_chn
Now I understand the answer should be no, but who can give me a brief conclusion?
• Jan 17th 2008, 07:18 PM
Jhevon
Quote:

Originally Posted by kevin_chn
Now I understand the answer should be no, but who can give me a brief conclusion?

go by the definitions:

an nxn matrix \$\displaystyle A\$ is similar to an nxn matrix \$\displaystyle B\$ if there exists an (invertible) nxn matrix \$\displaystyle P\$ such that:

\$\displaystyle P^{-1}AP = B\$

an nxn matrix \$\displaystyle A\$ is congruent to an nxn matrix \$\displaystyle B\$ if there exists an (invertible) nxn matrix \$\displaystyle P\$ such that:

\$\displaystyle P^TAP = B\$

where \$\displaystyle P^T\$ is the transpose of the matrix \$\displaystyle P\$

for similar matrices to be congruent as well, therefore, we must have \$\displaystyle P^{-1} = P^T\$, which will not always be true. you can find a counter-example to show this

see here for a counter-example