Recent content by linearishard

All I have is that if a singular value is the eigenvalue of ATA, then A must be positive semi definite or the signs will be different on at least one eigenvalue. I don't know where to start with symmetry or if my assumption is correct.

what do you mean by that? What is the forward and reverse directions?

Yeah I did that but it seemed too simple, my study guide says I should be using SVD. Is it actually unnecessary?

Hi, one more question! How do I prove that A has eigenvalues equal to its singular values iff it is symmetric positive definite? I think I have the positive definite down but I can't figure out the symmetric part. Thanks!

Hi, I have another question, if A and B are mxn matrices, how do I prove that $AA^T = BB^T$ iff $A = BO$ where $O$ is some orthogonal matrix? I think I need to use a singular value decomposition but I am not sure. Thanks!

Thank you for your response, could you please explain to me the logic behind the last line? How do you go from <ATAui,uj> to <Aui,Auj>?

Hi, yes that is what I meant! Sorry!

Hi! I have an orthonormal basis for vector space $V$, $\{u_1, u_2, ..., u_n\}$. If $(v_1, v_2, ..., v_n) = (u_1, u_2, ... u_n)A$ where $A$ is a real $n\times n$ matrix, how do I prove that $(v_1, v_2, ... v_n)$ is an orthonormal basis if and only if $A$ is an orthogonal matrix? Thanks!