Help please. linear algebra university questions.

• Nov 21st 2009, 05:49 AM
rebecca2812
Help please. linear algebra university questions.
Linear algebra question

Let A be a n x n symmetric matrix, and let L be an eigenvalue of A with corresponding eigenvector X.

Show that L is also an eigenvalue of the matrix P(transposed)AP where P is an orthogonal matrix. State the corresponding eigenvector of P(transposed)AP.

How is the result modified if P(transposed)AP is a diagonal matrix D?

I have no idea even where to start. Thanks guys.
• Nov 21st 2009, 07:08 AM
Shanks
By using the definition of eigenvalue and the property of orthogonal matrix, It is easy to prove that L is the eigenvalue of the matrix P(transposed)AP when P is an orthogonal matrix(note that P(transposed)P=E).
the corresponding eigenvector of P(transposed)AP is P(transposed)X.
If P(transposed)AP is a diagonal matrix D,then D gives all the eigenvalue of A, and the corresponding eigenvector of each eigenvalue coincide with the corresponding column vectors of P.
• Nov 21st 2009, 09:17 AM
tonio
Quote:

Originally Posted by rebecca2812
Linear algebra question

Let A be a n x n symmetric matrix, and let L be an eigenvalue of A with corresponding eigenvector X.

Show that L is also an eigenvalue of the matrix P(transposed)AP where P is an orthogonal matrix. State the corresponding eigenvector of P(transposed)AP.

How is the result modified if P(transposed)AP is a diagonal matrix D?

I have no idea even where to start. Thanks guys.

1) As $P$ is a orthogonal matrix it is invertible and in fact $P^t=P^{-1}$ .

2) Being P invertible there exists a vector $u$ s.t. $Pu=x\Longleftrightarrow u=P^{-1}x$ .

Well, now using that all the maps here are linear, show that $u$ is an eigenvector of $P^tAP$ corresponding to the eigenvalue $\lambda$
• Nov 21st 2009, 11:10 AM
rebecca2812
I still don't understand the proof for the first part. I'm never going to pass this module, I can't even understand it when people spell it out to me. I hate matrices.:(
• Nov 21st 2009, 11:27 AM
tonio
Quote:

Originally Posted by rebecca2812
I still don't understand the proof for the first part. I'm never going to pass this module, I can't even understand it when people spell it out to me. I hate matrices.:(

Well, that "optimistic' attitude won't make a lot for you, either. You've plenty of books, internet resources, etc. to try to understand.

Tonio
• Nov 22nd 2009, 05:20 AM
HallsofIvy
Quote:

Originally Posted by rebecca2812
Linear algebra question

Let A be a n x n symmetric matrix, and let L be an eigenvalue of A with corresponding eigenvector X.

Show that L is also an eigenvalue of the matrix P(transposed)AP where P is an orthogonal matrix. State the corresponding eigenvector of P(transposed)AP.

How is the result modified if P(transposed)AP is a diagonal matrix D?

I have no idea even where to start. Thanks guys.

You know that AX= LX. Let $Y= P^TX= P^{-1}X$ Then $X= PY$ and so AX= A(PY)= APY= LX. Now Apply $P^T$ to both sides of APY= LX.
• Nov 22nd 2009, 11:04 AM
rebecca2812
thanks so much, that was what I needed, have done my whole cw now. cheers everyone.