$UDU^{T}$ factorization of a matrix

Question

So, there is this question in which I am suppose to find $UDU^{T}$ factorization of $$A = \begin{bmatrix} -3&1&-1\\1&-3&1\\-1&1&-3 \end{bmatrix}$$ I don't understand what is the meaning of $UDU^{T}$ factorization. I don't want solution as such. I just want to know what the question is asking. Any help is appreciated.

EDIT

I just know that $xAx^{T}$ here represents a standard quadratic form. Is it related to something like that?

Answer 1

Are you familiar with the eigendecomposition?

$D$ will be your diagonal matrix with the eigenvalues in the diagonal and $U$ is your matrix of eigenvectors normalized so that each eigenvector has norm 1.

Answer 2

It is in fact the same prominent Schur decomposition for square matrices. Check the link below for further reading:

https://en.wikipedia.org/wiki/Schur_decomposition