I think eigenvalue product corresponding eigenvector has same effect as the matrix product eigenvector geometrically. I think my former understanding may be too naive so that I cannot find the link …

The eigenvalues of an orthogonal matrix needs to have modulus one. If the eigenvalues happen to be real, then they are forced to be $\pm 1$. Otherwise though, they are free to lie anywhere on the unit …

There are already good answers about importance of eigenvalues / eigenvectors, such as this question and some others, as well as this Wikipedia article. I know the theory and these examples, but n...

Think 'eigenspace' rather than a single eigenvector when you have repeated (non-degenerate) eigenvalues. – copper.hat To put the same thing into slightly different words: what you have here is a …

I'm studying eigenvector and eigenvalue but there are some confusing things to me. (1) Eigenvectors are not unique (2) If eigenvectors come from distinct eigenvalues, then eigenvectors are unique.

Edit: If $A$ has $n$ distinct eigenvalues then $A$ is diagonalizable (because it has a basis of eigenvalues). Two diagonal matrices with the same eigenvalues are similar and so $A$ and $B$ are …

I am trying to prove some statements about singular value decomposition, but I am not sure what the difference between singular value and eigenvalue is. Is "singular value" just another name for

Proposition: Given an endomorphism $A$ of a finite dimensional vector space $V$ equipped with a nondegenerate bilinear form $\langle\cdot,\cdot\rangle$, the endomorphisms $A$ and $A^*$ have …

It's a simple exercise to show that two similar matrices has the same eigenvalues and eigenvectors (my favorite way is noting that they represent the same linear transformation in different bases).

The quadratic formula is actually correct on the Harvard site. It's just a different way of writing it.