Similar matrix proof involving idempotency

[jawhnay]

Homework Statement

Prove that if A is idempotent matrix of order n and B is similar to A, then B is idempotent

The Attempt at a Solution

I was just wondering if my attempt at the solution right here is correct.

Given: A²=A and B=Q^-1AQ

Attempt: B=Q^-1QA B=IA B=A

Therefore, B=A^2 B=A(A) B=B(B) B=B²

 

[Dick]

Homework Statement

Prove that if A is idempotent matrix of order n and B is similar to A, then B is idempotent

The Attempt at a Solution

I was just wondering if my attempt at the solution right here is correct.

Given: A²=A and B=Q^-1AQ

Attempt: B=Q^-1QA B=IA B=A

Therefore, B=A^2 B=A(A) B=B(B) B=B²

No, not right. Matrices don't necessarily commute. You can't just change AQ into QA. Just work out (Q^(-1)AQ)^2.

 

[Mark44]

Homework Statement

Prove that if A is idempotent matrix of order n and B is similar to A, then B is idempotent

The Attempt at a Solution

I was just wondering if my attempt at the solution right here is correct.

Given: A²=A and B=Q^-1AQ

The problem says that A is idempotent of order n, not 2.

Attempt: B=Q^-1QA B=IA B=A

You can't do this (switch the order of A and Q). Matrix multiplication is generally not commutative.

Start with B² and show that it equals B.

For the real problem, you'll probably need to use induction.

Therefore, B=A^2 B=A(A) B=B(B) B=B²

 

[jawhnay]

If it says that B is similar to A, then am I allowed to assume A is similar to B a well?

 

[Mark44]

It's easy to show, but you don't need to do that. Dick and I have both suggested the same thing. Why don't you start with that?

 

[jawhnay]

I know how to do it, but I'm just wondering if I can assume that if I'm only given that B is similar to A.

 

[Dick]

I know how to do it, but I'm just wondering if I can assume that if I'm only given that B is similar to A.

Well, can you show that if B is similar to A then A is similar to B?

 

[jawhnay]

B=q^-1aq
qb=q^-1qaq
q^-1bq=q^-1qaq^-1q
q^-1bq=iia
q^-1bq=a

the letters are not showing up in caps for some reason.

 

[Dick]

B=q^-1aq
qb=q^-1qaq
q^-1bq=q^-1qaq^-1q
q^-1bq=iia
q^-1bq=a

the letters are not showing up in caps for some reason.

I wouldn't worry about the caps. But you are juggling the order of the matrices around again to force things to work the way you want them to. They don't commute. Try that again without assuming the matrices commute.

 

[Mark44]

B=q^-1aq
qb=q^-1qaq

You can't do this (above). On the left side of the equation, you multiplied on the left by Q, but on the right side of the equation, you multiplied on the right. When you multiply both sides of an equation by a matrix, you have to multiply both sides in the same way - either both sides on the left or both sides on the right.

q^-1bq=q^-1qaq^-1q
q^-1bq=iia
q^-1bq=a

the letters are not showing up in caps for some reason.

You probably hit your CAPS LOCK key.