How do I know if a matrix in quadratic form, e.g. D'MD is positive or negative (semi)definite? M here is the residual maker matrix for X, so I know that it is symmetric. I know what the definitions are for postive/negative (semi)definite but I'm unsure on how to tell if the matrix is positive or negative.
In my case, I have the model $y=x_i{\beta}+z_i{\delta}+{\epsilon}_i$ and my quadratic form comes from calculating the expectation of standard deviation when the $z_i{\delta}$ term is omitted. I actually have $(z{\delta})'M_x(z{\delta})/(n-k)$, which I know is in quadratic form but I'm unsure on how to work it out.
I've been told that it's positive but with no reasons.
Oh, just done another similar question where I had C$M_x$C', is the following correct: $CM_xC'=CM_xM_xC'$ since $M_x$ is idempotent
$CM_xM_xC'=CM_xM_x'C'$ since $M_x$ is symmetric
$CM_xM_x'C'=DD'$ where $D=CM_x$
So $DD'$ is positive semidefinite since it equals ${\sum_{i=1}^n}d_i^2$ ?
A quadratic form associated to a symmetric matrix is positive definite (resp. semi-positive, negative definite, semi-negative) if its eigenvalues are all $\,>0\,$ (resp. $\ge 0$, $<0$, $\le 0$).
See also the notion of
signatureof a quadratic form, in Gauß's decomposition as a sum of squares of linear forms: the number of squares with a positive coefficient is equal to the number of positive eigenvalues and similarly for the number of squares with a negative coefficients, since one can build an orthonormal basis of eigenvectors.