Eigenvalues of the product of two symmetric matrices

Question

This is mostly a reference request, as this must be well-known!

Let $A$ and $B$ be two real symmetric matrices, one of which is positive definite. Then it is easy to see that the product $AB$ (or $BA$, which has the same eigenvalues) is similar to a symmetric matrix, so has real eigenvalues. Take the vectors of eigenvalues of $A$ and of $B$, sorted in decreasing order, and let their componentwise product be $ab$. What is known about the relationship (e.g., inequalities) between $ab$ and the vector of eigenvalues of the product $AB$ (also taken in decreasing order)?

Some experimentation gives the conjecture that there is a majorization order between them, for instance. This must be well-known!

Answer 1

Here are the results that you are probably looking for.

The first one is for positive definite matrices only (the theorem cited below fixes a typo in the original, in that the correct version uses $\prec_w$ instead of $\prec$).

Theorem (Prob.III.6.14; Matrix Analysis, Bhatia 1997). Let $A$ and $B$ be Hermitian positive definite. Let $\lambda^\downarrow(X)$ denote the vector of eigenvalues of $X$ in decreasing order; define $\lambda^\uparrow(X)$ likewise. Then, \begin{equation*} \lambda^\downarrow(A) \cdot \lambda^\uparrow(B) \prec_w \lambda(AB) \prec_w \lambda^\downarrow(A) \cdot \lambda^\downarrow(B), \end{equation*}

where $x \cdot y := (x_1y_1,\ldots ,x_ny_n)$ for $x,y \in \mathbb{R}^n$ and $\prec_w$ is the weak majorization preorder.

However, when dealing with matrix products, it is more natural to consider singular values rather than eigenvalues.

Therefore, the relation that you might be looking for is the log-majorization \begin{equation*} \log \sigma^\downarrow(A) + \log\sigma^\uparrow(B) \prec \log\sigma(AB) \prec \log\sigma^\downarrow(A) + \log\sigma^\downarrow(B), \end{equation*} where $A$ and $B$ are arbitrary matrices, and $\sigma(\cdot)$ denotes the singular value map.

Reference

1. R. Bhatia. Matrix Analysis. Springer, GTM 169. 1997.

Answer 2

Let $A,B$ be $n \times n$ Hermitian matrices whose eigen values are non-negatives. Let $ \lambda_1(X) \geq \lambda_2(X) \geq \cdots \geq \lambda_n(X)$ denote ordered eigenvalues of $n \times n$ Hermitian matrix $X$.

Then, $\lambda_j(AB) \leq \lambda_i(A)\lambda_{j-i+1}(B),$ for any $i \leq j$.

Proof. Let $a_i,b_i$ and $c_i$ be the unitary eigenvectors of $A,B$ and $A+B$, respectively. Namely, $A a_i = \lambda_i(A)a_i$ and so on.

Let $V_a,V_b$ and $V_c$ are vector spaces spanned by the vectors $\{a_i,...,a_n\}$ and $\{b_{j-i+1},...,b_n\}$ and $\{c_1,...,c_j\}$, respectively. Then, $V_a \cap V_b \cap V_c \neq 0$. In fact, for arbitrary vector spaces $X,Y$, we can say $\dim (X \cap Y) = \dim(X)+\dim(X) -\dim(X+Y )$ and using this twice, it follows that

$\dim(V_a \cap V_b \cap V_c ) =\dim(V_a \cap( V_b \cap V_c) )$

$=\dim(V_a) + \dim( V_b \cap V_c) - \dim(V_a + ( V_b \cap V_c)) $

$=\dim(V_a) + \dim( V_b ) + \dim( V_c) - \dim( V_b +V_c) - \dim(V_a + ( V_b \cap V_c)) $

$\geq\dim(V_a) + \dim( V_b ) + \dim( V_c) - n -n $

$= (n-i+1) + (n-j+i-1)+1 + j - n -n$

$= 1,$

and thus there is a nonzero unit vector $x \in V_a \cap V_b \cap V_c.$

Because $x \in V_c$, we can write $x= \sum_{\nu =1}^j x^\nu c_\nu$ and then,

$<x,(AB)x>= < \sum_{\nu =1}^j x^\nu c_\nu , (AB) \sum_{\nu =1}^j x^\nu c_\nu >$

$ = < \sum_{\nu =1}^j x^\nu c_\nu , \sum_{\nu =1}^j x^\nu \lambda_\nu(AB)c_\nu >$

$ = \sum_{\nu =1}^j | x^\nu |^2 \lambda_\nu(AB)$

$ \geq \sum_{\nu =1}^j | x^\nu |^2 \lambda_j(AB)$

$ = \lambda_j(AB) \cdots \cdots \cdots (1).$

Similarly, because $x \in V_a$, we can write $x= \sum_{\nu =i}^n x^\nu a_\nu$ and then,

$<x,Ax>= < \sum_{\nu =i}^n x^\nu a_\nu , A\sum_{\nu =i}^n x^\nu a_\nu >$

$ = < \sum_{\nu =i}^n x^\nu a_\nu , A\sum_{\nu =i}^n x^\nu a_\nu >$

$ = \sum_{\nu =i}^n | x^\nu |^2 \lambda_\nu(A )$

$ \leq \sum_{\nu =i}^j | x^\nu |^2 \lambda_i(A )$

$ = \lambda_i(A ) \cdots \cdots \cdots (2).$

Similarly, $<x,Bx> \leq \lambda_{j-i+1}(B ) \cdots \cdots \cdots (3).$

Combining the inequalities (1),(2),(3) and Cauchy Schwartz inequality, we get

$\lambda_j(AB) \leq <x,(AB)x> \leq |<A^*x , Bx>| \leq |A^*x| \dot | Bx| \leq \lambda_i(A ) \lambda_{j-i+1}(B ). $

This completes the proof.