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Let $A,B$ be $n\times n$ real matrices such that $A^2$ is positive-definite and $B$ is symmetric. Show that $A^2-B$ is positive-definite if and only if $A^{-1}BA^{-1}$ has all eigenvalues $<1$.

If we choose invertible matrix $P$ such that $P'(A^2)P=I_n$, then $A^2-B$ is positive-definite implies $P'(A^2-B)P=I-P'BP$ is positive-definite, and eigenvalues of $P'BP$ is $<1$. But it is inconsistent with $|P' A(x I-A^{-1}BA^{-1})AP|=|x P'P-P'BP|$. If $P$ is orthogonal, OK then. However, ...

Semiclassical
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xldd
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    One point you write does seem suggestive: $A^2-B=A(I_n-A^{-1} BA^{-1})A$. Maybe consider the implication of $A^{-1}B A^{-1}$ having an eigenvalue $\geq 1$? – Semiclassical Jan 20 '21 at 06:32
  • what does $A^2$ is positive definite mean? If that allows for non-symmetric $A^2$ then what does $A^{-1}BA^{-1}$ having all eigs $\lt 1$ mean? (i.e. for non-symmetric $A^2$ and e.g. $B:=I$ then there is no reason for $A^{-2}$ to have entirely real spectrum) – user8675309 Jan 20 '21 at 07:45
  • @user8675309 $A^2$ is symmetric, and positive-definiete, but $A$ is perhaps not. – xldd Jan 20 '21 at 08:25

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If $P$ is invertible and $H$ is Hermitian, the matrix product $P^\ast HP$ is said to be congruent to $H$. Positive or negative (semi)definiteness are preserved under congruence. In fact, since $P$ is invertible, $v=Pu$ is a one-to-one correspondence between two vectors $u$ and $v$. Therefore $v^\ast Hv>0$ if and only if $u^\ast P^\ast HPu>0$, and likewise when $>$ is replaced by $\ge,<$ or $\le$.

In your case, let $P=(A^2)^{1/2}$. Then $A^2-B=P^2-B=P(I-P^{-1}BP^{-1})P$ is positive definite if and only if $I-P^{-1}BP^{-1}$ is positive definite, and the latter occurs if and only if all eigenvalues of $P^{-1}BP^{-1}$ are less than $1$.

Now, in general, if $X$ and $Y$ are two square matrices of the same sizes, then $XY$ and $YX$ have the same characteristic polynomials and hence also the same spectra. Therefore $P^{-1}BP^{-1},\,P^{-2}B=A^{-2}B$ and $A^{-1}BA^{-1}$ have the same eigenvalues. Hence the result follows.

user1551
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