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Let $P$ and $Q$ be positive definite matrices. Consider the following matrix equation $$\label{star}\tag{$\star$} XPX^\top - P = -Q, \quad X\in\mathbb{R}^{n\times n}. $$

My question. Is it true that any solution of \eqref{star} can be written as $X=(P-Q)^{1/2}TP^{-1/2}$ with $T$ being an arbitrary orthogonal matrix?

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    $\begingroup$ As (1.) an straightforward computation shows that if $X$ is a solution then $A={(P-Q)}^{-1/2}XP^{1/2}$ satisfies $AA^T=Id$.However we need to add the assumption $Q<P$ $\endgroup$ – Ali Taghavi May 18 '18 at 9:28
  • $\begingroup$ Have you tested (2) on at least, say, 10000 examples before asking here? $\endgroup$ – Federico Poloni May 18 '18 at 10:11
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    $\begingroup$ @FedericoPoloni: I did run some numerical simulations in matlab. However, after posting this question, I found out that there was a bug in the generation of a random orthogonal matrix. Once fixed this, counterexamples can be found. I will now edit the OP. $\endgroup$ – Ludwig May 18 '18 at 16:11
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    $\begingroup$ @AliTaghavi It would be useful if you posted your comment as an answer, so that this question can be marked as answered/accepted. You don't need much more detail. $\endgroup$ – Federico Poloni May 19 '18 at 8:20
  • $\begingroup$ @FedericoPoloni Thanks for your suggestion. I wrote an answer to the previous version of the question. $\endgroup$ – Ali Taghavi May 20 '18 at 8:03
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I am considering the previous version of your question which contained two parts.

Part 1) We assume $Q<P$ which means that $P-Q$ is a strictly positive matrix. Of course this condition is a necessary condition for the equation to have a solution. So we assume $P, Q,P-Q$ are positive invertible matrices.

First note that the equation has always a solution. Here is a proof:

WLOG we may assume that the positive matrix $P-Q$ is a diagonal matrix. Because every positive matrix is orthonormally diagonalizable. We consider the inner product whose tensor matrix is $P$. Now a solution to your matrix equation corresponds to a basis consistong of mutually orthogonal vectors whose norms are not necessarilly unit but are determined by the entries of the diagonal matrix $P-Q$. Now one can show that if $X$ is a solution of the equation in your question then $A={(P-Q)}^{-1/2}XP^{1/2}$ satisfies $AA^T=Id$

Part 2) The answer is yes if we further assume that $PQ=QP$.

Note that the spectrum of the solution $X={(P-Q)}^{1/2}T P^{-1/2}$ is equal to the spectrum of $T{(P-Q)}^{1/2}P^{-1/2}$ because $AB$ and $BA$ have the same spectrum. Now the commutativity assumption $PQ=QP$ implies that $H={(P-Q)}^{1/2}P^{-1/2}$ is a positive matrix.

Now we use the following lemma:

Lemma: If $T$ is an orthonormal matrix and $H$ is a positive matrix Then $\lambda_{min} (TH)= \lambda_{min}(H)$ and $\lambda_{Max} (TH)=\lambda_{Max}(H)$.

Proof: It is sufficient to prove the following statement. Then the lemma is a consequence of a simple rescaling and consideration of the fact that "The spectrum of the inverse is equal to the inverse of the spectrum"

Statement: If all eigenvaluse of a positive matrix $H$ lie in the interior of the unit disc of the complex plane then the same is true for all eigenvalues of $TH$ where $T$ is an arbitrary orthonormal matrix:

Proof of the Statement: WLOG we may assume that $H$ is a diagonal matrix because every positive matrix is unitary equivalent to a diagonal matrix. For diagonal matrices the statement is obvious. Because if $THV=\lambda V$ for some $\lambda \in \mathbb{C}$ with $|\lambda|>1$ and $V \in \mathbb{C}^n$ then $|HV|>|V|$ which is impossible since all entries of diagonal matrix $H$ are positive element less than $1$.

The second part of the previous version of your question is a motivation to consider the following question in the context of complex $C^*$ algebras. We keep the same notations $\lambda_{min}$ and $\lambda_{Max}$ for the case of $C^*$ algebras.

Question : let $A$ be a simple $C^*$ algebra and $u$ be a unitary element with the property that for every two positive element $a,b$ we have

$$\lambda_{min} (aub)=\lambda_{min}(ab)\\ \lambda_{Max}(aub)=\lambda_{Max}(ab)$$

Does this imply that $u$ is an scalar element?

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    $\begingroup$ I can't follow this "the spectrum of the solution $X={(P-Q)}^{1/2}T P^{-1/2}$ is equal to the spectrum of $T{(P-Q)}^{1/2}P^{-1/2}$ because $AB$ and $BA$ have the same spectrum". The later statement implies just that the spectrum of the product of some matrices is invariant under cyclic permutation, not every permutation. am I missing something? $\endgroup$ – Mahdi May 20 '18 at 20:27
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    $\begingroup$ @Mahdi Thanks for your comment. Yes you are completely right. the spectrum is invariant under cyclic permutation. But our extra assumption $PQ=QP$ implies that the permutation I used is actualy a cyclic permutation. Right? $\endgroup$ – Ali Taghavi May 20 '18 at 20:53
  • $\begingroup$ Yes, that is right. Thanks for the response. $\endgroup$ – Mahdi May 20 '18 at 23:57

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