Thread: Square matrix problems, need help :)

Question 1: Let A be a non-square matrix, say m x n for m is different from n, Then C = $\displaystyle AA^t$ and $\displaystyle D = A^tA $ are both matrices, but of different sizes.
Show that C and D both have the same non-zero eigenvalues. [Hint: write down the equation that must be satisfied by a vector v to be an eigenvalue $\displaystyle \lambda$, and try to adjust that equation so as to find an eigenvector of D with the same eigenvalue. you will need to explain why your proposed eigenvector is not O.]

Question 2: Is there a 3 x 3 symmetric matrix with eigenvalues $\displaystyle \lambda_{1} = 3, \lambda_{2} = 7, \lambda_{3} = 1 $ and corresponding eigenvectors $\displaystyle v_{1} = (1,0,0), v_{2} = (0,1,1), v_{3} = (0,1,-1)? $ Either find the matrix or explain why it can not exist.

What kind of help do you need? If you would show some attempt at these problems, we could see better where you are having problems and what kinds of hints would help you.

Originally Posted by HallsofIvy

What kind of help do you need? If you would show some attempt at these problems, we could see better where you are having problems and what kinds of hints would help you.

sorry, I have no idea where to start, even the hint wouldn't help. Can you please at least tell me which steps I should take that would be much appreciated