TITLE: Positive semi-definite of a matrix composed of semi-definite blocks
QUESTION [7 upvotes]: Say a matrix A is positive semi-definite. Let B be a square matrix composed of replicas of A as sub-blocks, s.t. $$B=\begin{pmatrix} A & A \\ A & A \\ \end{pmatrix},$$ or $$\begin{pmatrix} A & A & A \\  A & A & A \\ A & A & A \\ \end{pmatrix},$$ etc. Would $B$ be semi-definite as well?

REPLY [2 votes]: Let $X$ and $Y$ be symmetric positive semidefinite matrices. Then $X\otimes Y$, where $\otimes$ denotes the Kronecker product, is symmetric positive semidefinite as well.

Let
$$
X=E, \quad Y=A,
$$
where $E$ is a square matrix of ones (for the matrices in question, $E$ is $2\times 2$ or $3\times 3$). Note that $E=ee^T$ with $e=[1,1,\ldots,1]^T$, so $E$ is obviously symmetric positive semidefinite. Now use the fact above.