For a positive definite $n \times n$ matrix $A$ and a vector $x$ how to write the expression $Ax$ in terms of the eigenvalues of $A$

Question

For a positive definite $n \times n$ matrix $A$ and a vector $x$ I want to write the expression $Ax$ in terms of the eigenvalues of $A$. How to do that ?

Answer 1

If $A$ is symmetric positive definite, then the eigenvectors form an orthogonal basis and $A$ can be written as:

$$A = \sum_{i=1}^{n}{\lambda_i u_i u_i^T}$$

Where the $\lambda_i$ are the eigenvalues (real positive here) and the $u_i$ are the eigenvectors.

Then, for any input $x$

$$Ax = \sum_{i=1}^{n}{\lambda_i (u_i^Tx_i) u_i} $$