# Questions tagged [matrix-equations]

This tag is for questions related to equations, with matrices as coefficients and unknowns. A matrix equation is an equation in which a variable stands for a matrix .

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### A conjecture on the Lyapunov equation

Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e., all the eigenvalues of $A$ have strictly negative real part). Let $X\in\mathbb{R}^{n\times n}$ be a positive semi-definite matrix of ...
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### Simplifying Trace of a Matrix Expression ${\rm Tr} \left( (I- D^{-1} BA^2) B (I- D^{-1} BA^2)^T \right)$

Let $B$ be a symmetric invertible matrix and let $A$ be a diagonal matrix with non-zero entries on the main diagonal. I am trying to simplify the following expression \begin{align} {\rm Tr} \left( ...
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### Spectral norm inequality: Applying lipschitz continous function elementwise

Let $A, B$ be some $m\times n$ matrices in $\mathbb{R}$ and $f:\mathbb{R} \to \mathbb{R}$ a Lipschitz continuous function, i.e. $\|f(x) - f(y)\|_2 \leq L\|x-y\|_2$ for some $L$. For the Frobenius norm ...
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### Convex conjugate of a matrix function

Let $S$ be some (not necessarily convex) subset of positive semidefinite matrices. What is the convex conjugate of the function $$f(x) = \sup_M \left\{ x^T M x \;:\; M \in S \right\},$$ that is, ...
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### (Nonunique) Solvability of Sylvester Equation

I am interested in stating existence of solution of a Sylvester equation $$AX - XB = C,$$ where $A$, $B$, $C$, and $X$ are $(n,n)$ matrices. Existence of a unique solution $X$ is given, if $A$ ...
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### Matrix equation $A=PBP^{-1}$

Suppose $P$ is invertible and $A=PBP^{-1}$ . Solve for $B$ in terms of $A$. My attempt: I just left multiplied the equation by $P^{-1}$ and right multiplied it by $P$ so that I got $B=P^{-1}AP$. Is ...
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### Intrinsic camera parameter from three vanishing points

I would like to recover the intrinsic parameter from three vanishing points of a digital image. I read a bit on the famous textbook Multiple View Geometry in Computer Vision (pag. 226) but I couldn't ...
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### An abstraction of matrices that can be added, inverted, transposed, multiplied

$$x^TAx+x^Tb+c=(x-h)^TA(x-h)+k$$ where $$h=-(A+A^T)^{-1}b$$ $$k=c-h^TAh$$ Determining $h$ and $k$ above is called "completing the square" and requires matrix addition, inversion, transposition, ...
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### Limitations of Bruck-Ryser proof method

I'm reading (the rather shrot paper) The nonexistence of certain finite projective planes by Bruck and Ryser, in which they prove that $\quad$ If $N$ is $1$ or $2$ modulo $4$, and if the square free ...
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### Finding matrix $\mathbf{Q}$ that satisfies $\mathbf{J=Q^TAQ}$ of skew-symmetric $\mathbf{A}$ and $\mathbf{J}$

Let $\mathbf{A}$ and $\mathbf{J}$ be real, full-rank, skew-symmetric, square matrices. If we know those matrices, can we find a real square matrix $\mathbf{Q}$ that satisfies $\mathbf{J=Q^TAQ}$? ...
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### Uniform Smoothness Inequality for Schatten Norms

I'm looking for a proof of the inequality $$\left[ \frac12(\left\|A+B\right\|_p^p + \left\|A-B\right\|_p^p)\right]^{2/p} \leq \left\|A\right\|_p^2 +C_p \left\|B\right\|_p^2$$ where A and B are ...
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### How calculate the singular values of the Neumann series $M=\sum_{n=1}^\infty A^n$?

Given a matrix $A$, the Neumann series https://en.wikipedia.org/wiki/Neumann_series is defined as $$M=\sum_{n=0}^\infty A^n.$$ For a converge $M$, how to calculate the singular value? Is it possible ...
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### About the characteristic equation of a square matrix (Cayley-Hamilton theorem)

Take, for example, $A$ of order $2 \times 2$. The characteristic equation comes to be $$A^2 - (\text{tr}(A)) A + \det (A) = 0$$ But is this the only quadratic equation that this matrix satisfies? ...
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Let $$G=\left\{ \begin{bmatrix} \bar{a} & \bar{b} \\ \bar{0} & \bar{c} \end{bmatrix} \text{with \bar{a} and \bar{c} in \mathbb{F}^{*}_{7} and \bar{b} in \mathbb{F}_{7}}\... • 901 3 votes 0 answers 251 views ### Inverse of element-wise positive matrices inequality Consider two matrices A and B that satisfy the inequality A \leq B (the relation \leq is understood element-wise). The matrices A and B are positive (there elements are positive). Under which ... • 389 3 votes 2 answers 310 views ### What is the n-time iterated adjugate of an n\times n matrix A? What is \underbrace{\text{adj}\Big(\text{adj}\big(\ldots(\text{adj}}_{n\text{ adj}}\ A)\ldots\big)\Big), where \text{adj} is written n times, and the order of the matrix A is n\times n? Can ... 3 votes 0 answers 64 views ### Show that the values of the following determinants are not zero without actually finding the exact values$$\begin{bmatrix} 111 & 100 & 225 & 235\\ 220 & 312 & 220 & 410\\ 215 & 180 & 268 & 305\\ 315 & 145 & 205 & 122 \end{bmatrix}$$Guys is it enough to ... • 107 3 votes 0 answers 62 views ### How to find the existence criterion for real solution on the complex Vandermonde system Given postive integer M \in \mathbb{N}, find the condition on the the combination of distinct p integers$$i_{0}, i_{1}, \ldots, i_{p-1} \in \{0,1,\ldots,M-1 \}, \omega = e^{-j\frac{2\pi}{M}}$$... 3 votes 0 answers 203 views ### Computing the discrete-time difference equation when the system matrix is non-invertible If we have a continuous time equation,$$ \dot{x}(t) = A x(t) + B u(t)$$where A \in \mathbb{R}^{n \times n}, x \in \mathbb{R}^{n\times1}, B \in \mathbb{R}^{n \times m}, u \in \mathbb{R}^{m \times ... 3 votes 0 answers 44 views ### A set of four matrix equations Let \{\mathbf b_1,\mathbf b_2,\mathbf b_3\} be a basis for \mathbb R^3 and consider three arbitrary vectors \mathbf w_i=\sum_j\omega_{ij}\mathbf b_j. We define the following two 3\times 9 ... • 177 3 votes 0 answers 48 views ### Find matrix S such that diag(S)=d and S^{-1}_{ij}=P_{ij} \forall i \neq j I want to find a matrix S with a fixed diagonal d whose inverse matches P on all of its off-diagonal elements. You can assume S and P are positive definite, and d>0 (all elements of d... • 2,210 3 votes 0 answers 126 views ### Characterize set of matrices that are orthogonal in two particular senses Does there exist an analytical characterization of the set of matrices \Gamma_k\in\mathbb{R}^{m\times n} such that both$$ \sum_{k=1}^K\Gamma_k^T\Lambda_0\Gamma_k=I $$and$$ \sum_{k=1}^K\Gamma_k\...
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Let $A=\begin{bmatrix} 2 & 6 \\ 0 & 2 \end{bmatrix}$ and $X$ be a $2\text{x}2$ matrix with real entries. Solve the following equation: $$X^5+X=A$$ I solved this, but I feel like I got lucky. ...