linear algebra - Product of Positive Matrices

Lemma 1. Let $k$ be a field, and $Ain k^{ntimes n}$ be a symmetric matrix.

(a) Then, there exist an invertible matrix $Uin k^{ntimes n}$ and a diagonal matrix $Din k^{ntimes n}$ such that $A=U^TDU$.

(b) Let the field $k$ be ordered. The matrix $A$ is nonnegative-definite if and only if all entries of the matrix $D$ are nonnegative.

I'm writing a proof of this, mainly because too many sources do it wrong (for instance, by assuming $k$ to be ordered in (a), where it is useless, or using the spectral theorem, which is much stronger and requires $k=mathbb R$). But it isn't necessary for your question: you only seem to need the $k=mathbb R$ case, where any proof would do.

EDIT: I'm not writing a proof of this. Too tired from the rest. See Proposition 15.1 in J. S. Milne's "Algebraic Groups, Lie Groups, and their Arithmetic Subgroups" Chapter I for a proof of Lemma 1 (a), and derive Lemma 1 (b) from it.

Corollary 2. Let $k$ be an ordered field. Let $Ain k^{ntimes n}$ and $Bin k^{ntimes n}$ be two symmetric nonnegative-definite matrices. Then, $mathrm{Tr}left(ABright)geq 0$.

Proof of Corollary 2. Consider the Kronecker product $Aotimes Bin k^{n^2times n^2}$ of the two matrices $A$ and $B$. This Kronecker product $Aotimes B$ is defined as the matrix $left(A_{i,j}B_{i',j'}right)_{left(1,1right)leq left(i,i'right)leq left(n,nright), left(1,1right)leq left(j,j'right)leq left(n,nright)}$. Here, $A_{i,j}$ is the $left(i,jright)$-th entry of the matrix $A$, and $B_{i',j'}$ is the $left(i',j'right)$-th entry of the matrix $B$. Besides, the coordinates in the vector space $k^{n^2}$ are indexed by pairs $left(i,i'right)inleftlbrace 1,2,...,nrightrbrace^2$, and these pairs are ordered lexicographically.

Lemma 1 (a) yields the existence of an invertible matrix $Uin k^{ntimes n}$ and a diagonal matrix $Din k^{ntimes n}$ such that $A=U^TDU$, and Lemma 1 (b) shows that all entries of the matrix $D$ are nonnegative. Similarly, Lemma 1 (a) (applied to the matrix $B$ instead of $A$) yields the existence of an invertible matrix $Vin k^{ntimes n}$ and a diagonal matrix $Ein k^{ntimes n}$ such that $B=V^TEV$, and Lemma 1 (b) shows that all entries of the matrix $E$ are nonnegative. Thus, $Aotimes B=left(U^TDUright)otimesleft(V^TEVright)=left(Uotimes Vright)^Tleft(Dotimes Eright)left(Uotimes Vright)$, so that the matrix $Aotimes B$ is nonnegative-definite (because the matrix $Dotimes E$ is a diagonal matrix all of whose entries are nonnegative, and therefore it is nonnegative-definite).

Now, let $vin k^{n^2}$ be the vector given by $v_{left(i,i'right)}=left[i=i'right]$ for any pair $left(i,i'right)inleftlbrace 1,2,...,nrightrbrace^2$. Here, for any assertion $mathcal A$, we denote by $left[mathcal Aright]$ the truth value of $mathcal A$, defined by $left[mathcal Aright]=1$ if $mathcal A$ is true and $left[mathcal Aright]=0$ otherwise.

Now, an easy computation yields $v^Tleft(Aotimes Bright)v=mathrm{Tr}left(AB^Tright)$. Since $B^T=B$, this becomes $v^Tleft(Aotimes Bright)v=mathrm{Tr}left(ABright)$. But $v^Tleft(Aotimes Bright)vgeq 0$, since $Aotimes B$ is a nonnegative-definite matrix. Thus, $mathrm{Tr}left(ABright)geq 0$, proving Corollary 2.