Thread: Last Linear Algebra problem

Defenition: Let A and B be nxn matrices. Then A is said to be similar to B, denoted A is congruent to B, if there exists an invertible nxn matrix P such that B=(P^-1)AP.

a. Let A=I_n. Prove that, if A is congruent to B, then B=I_n

b. Let A be a nxn idempotent matrix. Prove that if A is congruent to B, then B is idempotent.

c. Is the analogue of part b above the nilpotent matrices true? Why or Why not?

Originally Posted by natewalker205

Defenition: Let A and B be nxn matrices. Then A is said to be similar to B, denoted A is congruent to B, if there exists an invertible nxn matrix P such that B=(P^-1)AP.

a. Let A=I_n. Prove that, if A is congruent to B, then B=I_n

b. Let A be a nxn idempotent matrix. Prove that if A is congruent to B, then B is idempotent.

c. Is the analogue of part b above the nilpotent matrices true? Why or Why not?

In all of these cases you just need to observe that as $\displaystyle A$ and $\displaystyle B$ are congruent there exists an invertible matrix $\displaystyle P$ such that:

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

Now use what you are told in a, b and c and the answers follow at once

RonL