I am reading the introduce of linear system and eigenvalues. There I read if there is a matrix $A$ and vector $x$, it could find a eigenvalue $\lambda$ such that $$Ax = \lambda x$$
I have a really big diagonal matrix D (nxn) (so I don't have to diagonalize it). I need to compute matrix-vector multiplication $Dx$, so based on the eigenvalue property, can I say that I can retain the same result by multiplying the vector $x$ and a number $\lambda$? However, I try to find the eigenvalue of the matrix $D$ with matlab, it does give me $n$ eigenvalues, so which one should I use to estimate $Dx$? Thanks.