Skip to main content
MathOverflow

Return to Question

deleted 223 characters in body
Source Link
Ludwig
Ludwig
  • 2.7k
  • 1
  • 14
  • 26

Let $P$ and $Q$ be positive definite matrices. Consider the following matrix equation $$\label{star}\tag{$\star$} XPX^\top - P = -Q, \quad X\in\mathbb{R}^{n\times n}. $$ Moreover, given a matrix $M\in\mathbb{R}^{n\times n}$ let $\sigma(M)$ denote the spectrum of $M$ and $\lambda_{\min}(M):=\min_{\lambda\in\sigma(M)} |\lambda| $, $\lambda_{\max}(M):=\max_{\lambda\in\sigma(M)} |\lambda|$.

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?

Let $P$ and $Q$ be positive definite matrices. Consider the following matrix equation $$\label{star}\tag{$\star$} XPX^\top - P = -Q, \quad X\in\mathbb{R}^{n\times n}. $$ Moreover, given a matrix $M\in\mathbb{R}^{n\times n}$ let $\sigma(M)$ denote the spectrum of $M$ and $\lambda_{\min}(M):=\min_{\lambda\in\sigma(M)} |\lambda| $, $\lambda_{\max}(M):=\max_{\lambda\in\sigma(M)} |\lambda|$.

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?

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?

deleted second question
Source Link
Ludwig
Ludwig
  • 2.7k
  • 1
  • 14
  • 26

Let $P$ and $Q$ be positive definite matrices. Consider the following matrix equation $$\label{star}\tag{$\star$} XPX^\top - P = -Q, \quad X\in\mathbb{R}^{n\times n}. $$ Moreover, given a matrix $M\in\mathbb{R}^{n\times n}$ let $\sigma(M)$ denote the spectrum of $M$ and $\lambda_{\min}(M):=\min_{\lambda\in\sigma(M)} |\lambda| $, $\lambda_{\max}(M):=\max_{\lambda\in\sigma(M)} |\lambda|$.

  1. 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?
  1. Is it true that the eigenvalues of any $X$ solving \eqref{star} have moduli contained in the interval $[\lambda_{\min}((P-Q)^{1/2}P^{-1/2}),\lambda_{\max}((P-Q)^{1/2}P^{-1/2})]$?

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?

Let $P$ and $Q$ be positive definite matrices. Consider the following matrix equation $$\label{star}\tag{$\star$} XPX^\top - P = -Q, \quad X\in\mathbb{R}^{n\times n}. $$ Moreover, given a matrix $M\in\mathbb{R}^{n\times n}$ let $\sigma(M)$ denote the spectrum of $M$ and $\lambda_{\min}(M):=\min_{\lambda\in\sigma(M)} |\lambda| $, $\lambda_{\max}(M):=\max_{\lambda\in\sigma(M)} |\lambda|$.

  1. 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?
  1. Is it true that the eigenvalues of any $X$ solving \eqref{star} have moduli contained in the interval $[\lambda_{\min}((P-Q)^{1/2}P^{-1/2}),\lambda_{\max}((P-Q)^{1/2}P^{-1/2})]$?

Let $P$ and $Q$ be positive definite matrices. Consider the following matrix equation $$\label{star}\tag{$\star$} XPX^\top - P = -Q, \quad X\in\mathbb{R}^{n\times n}. $$ Moreover, given a matrix $M\in\mathbb{R}^{n\times n}$ let $\sigma(M)$ denote the spectrum of $M$ and $\lambda_{\min}(M):=\min_{\lambda\in\sigma(M)} |\lambda| $, $\lambda_{\max}(M):=\max_{\lambda\in\sigma(M)} |\lambda|$.

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?

deleted 2 characters in body
Source Link
Ludwig
Ludwig
  • 2.7k
  • 1
  • 14
  • 26

Let $P$ and $Q$ be positive definite matrices. Consider the following matrix equation $$\label{star}\tag{$\star$} XPX^\top - P = -Q $$$$\label{star}\tag{$\star$} XPX^\top - P = -Q, \quad X\in\mathbb{R}^{n\times n}. $$ where $X$ is any square real matrix. MoreoverMoreover, given a matrix $M\in\mathbb{R}^{n\times n}$ let $\sigma(M)$ denote the spectrum of $M$ and $\lambda_{\min}(M):=\min_{\lambda\in\sigma(M)} |\lambda| $, $\lambda_{\max}(M):=\max_{\lambda\in\sigma(M)} |\lambda|$.

  1. CanIs 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?
  1. Is it true that the eigenvalues of any $X$ solving \eqref{star} have moduli contained in the interval $[\lambda_{\min}((P-Q)^{1/2}P^{-1/2}),\lambda_{\max}((P-Q)^{1/2}P^{-1/2})]$?

Let $P$ and $Q$ be positive definite matrices. Consider the following matrix equation $$\label{star}\tag{$\star$} XPX^\top - P = -Q $$ where $X$ is any square real matrix. Moreover, given a matrix $M\in\mathbb{R}^{n\times n}$ let $\sigma(M)$ denote the spectrum of $M$ and $\lambda_{\min}(M):=\min_{\lambda\in\sigma(M)} |\lambda| $, $\lambda_{\max}(M):=\max_{\lambda\in\sigma(M)} |\lambda|$.

  1. Can any solution of \eqref{star} be written as $X=(P-Q)^{1/2}TP^{-1/2}$ with $T$ being an arbitrary orthogonal matrix?
  1. Is it true that the eigenvalues of any $X$ solving \eqref{star} have moduli contained in the interval $[\lambda_{\min}((P-Q)^{1/2}P^{-1/2}),\lambda_{\max}((P-Q)^{1/2}P^{-1/2})]$?

Let $P$ and $Q$ be positive definite matrices. Consider the following matrix equation $$\label{star}\tag{$\star$} XPX^\top - P = -Q, \quad X\in\mathbb{R}^{n\times n}. $$ Moreover, given a matrix $M\in\mathbb{R}^{n\times n}$ let $\sigma(M)$ denote the spectrum of $M$ and $\lambda_{\min}(M):=\min_{\lambda\in\sigma(M)} |\lambda| $, $\lambda_{\max}(M):=\max_{\lambda\in\sigma(M)} |\lambda|$.

  1. 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?
  1. Is it true that the eigenvalues of any $X$ solving \eqref{star} have moduli contained in the interval $[\lambda_{\min}((P-Q)^{1/2}P^{-1/2}),\lambda_{\max}((P-Q)^{1/2}P^{-1/2})]$?
Source Link
Ludwig
Ludwig
  • 2.7k
  • 1
  • 14
  • 26