1. ## 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.

2. 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.

3. 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 $\displaystyle P$ is a orthogonal matrix it is invertible and in fact $\displaystyle P^t=P^{-1}$ .

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

Well, now using that all the maps here are linear, show that $\displaystyle u$ is an eigenvector of $\displaystyle P^tAP$ corresponding to the eigenvalue $\displaystyle \lambda$

4. 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.

5. 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

6. 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 $\displaystyle Y= P^TX= P^{-1}X$ Then $\displaystyle X= PY$ and so AX= A(PY)= APY= LX. Now Apply $\displaystyle P^T$ to both sides of APY= LX.

7. thanks so much, that was what I needed, have done my whole cw now. cheers everyone.