Positive semi-definiteness of a matrix comprising block matrices of special form

Question

I am interested in characterizing the positive semi-definiteness of a matrix in the form

(A B C; D E F; H I G), where each one the the blocks A..G is a 3X3 matrix whose components are the same, e.g., A=(a a a; a a a; a a a), B=(b b b; b b b; b b b;), etc. Does anyone have any ideas how to exploit this form to characterize PSD?

thanks,

Gal

Answer 1

The matrix in question can be written as $$ \pmatrix{a&b&c\\d&e&f\\g&h&i}\otimes \pmatrix{1&1&1\\1&1&1\\1&1&1}=:M\otimes E, $$ where $\otimes$ denotes the Kronecker product. There is a certain link between the eigenvalues of the Kronecker product and its arguments which states that the eigenvalues of $M\otimes E$ are the products of eigenvalues of $M$ and $E$. Consequently, $M\otimes E$ has nonnegative eigenvalues if and only if $M$ does (because the eigenvalues of $E$ are simple $3$ and $0$ with multiplicity $2$).