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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 ...
Ludwig's user avatar
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7 votes
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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( ...
Boby's user avatar
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6 votes
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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 ...
danwin AI's user avatar
6 votes
0 answers
721 views

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, ...
BDD's user avatar
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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 $A$ ...
Jan's user avatar
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5 votes
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64 views

Group Structure of the $2\times2$ Matrix Sphere

I define the $2\times2$ matrix sphere as the quotient set $$\boxed{\mathbb{S}_{2}:=\{(X,Y)\in M_2(\mathbb{R})^2:X^2+Y^2=I\}/\sim}$$ where $M_2(\mathbb{R})^2=\mathbb{R}^{2\times2}\times\mathbb{R}^{2\...
K. Makabre's user avatar
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689 views

When does $\| AB \| =\| A \| \| B\| $ hold?

A matrix norm that satisfies $\| AB\| \leq \| A\| \| B \| $ is also a submultiplicative norm, so, if $A,B$ are both square matrices, when does the equality hold?
charesp's user avatar
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5 votes
1 answer
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Prove that if $AA^T=A^TA$ and $AB=BA$ then $AB^T=B^TA$

Prove that if $AA^T=A^TA$ and $AB=BA$ then $AB^T=B^TA$ where $A$ and $B$ are matrices. It doesn't say of what order they are. Can somebody help me with this exam problem? I know I should put my work ...
J.Dane's user avatar
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5 votes
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What is the probability of exactly one negative solution in a Fibonacci system of equations?

The Fibonacci numbers denoted by $F_i$ for $i\ge1$ are $$1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,\cdots$$ where they satisfy the property $F_{i+2}=F_{i+1}+F_i$. I have listed the first $15$ ...
TheSimpliFire's user avatar
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Convergence of iteration scheme of solving matrix equations

Consider the equation $A\mathbf{x}=\mathbf{b}$. It is equivalent to $$S\mathbf{x}=(S-A)\mathbf{x}+\mathbf{b}$$ where $S$ is a splitting matrix. We now consider the iteration scheme $$S\mathbf{x}_{i+1}=...
Nighty's user avatar
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5 votes
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Determinant of a function

I was thinking about matrices and then why arent there matrices with uncountable many values? (Probably this conecpt already exists for a very long time, but i don't know it) Assume there are ...
Kevin Meier's user avatar
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Can we neglect matrices with smaller eigenvalues in comparison to ones with larger eigenvalues?

Can two matrices be compared as being "small" and "large"? For example, consider matrix $X=X(t)$ as a function of parameter $t$ (say for time), such that \begin{equation} \frac{d ...
User101's user avatar
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Image of the map $f : M_n \to M_n$ given by $f(X) = X + \|X-I\|I$.

Let $M_n = M_n(\Bbb{C})$ be the algebra of $n\times n$ complex matrices and let $\|\cdot\|$ be the operator norm. Consider the map $$f : M_n \to M_n, \qquad f(X) = X + \|X-I\|I.$$ I'm interested in ...
mechanodroid's user avatar
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4 votes
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Matrix Recurence Relation, $M_n=M_{n-1}M_{n-2}$

Let $M_0, M_1\in\mathbb R^{k\times k}$ be matrices, and define $M_n=M_{n-1}M_{n-2}$. Then $M_n$ is a product of $F_n$ ($n$th Fibonacci number) many copies of $M_0$ or $M_1$. How do I compute the limit ...
Thomas Ahle's user avatar
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4 votes
1 answer
224 views

Matrix Equation $A^* B A = C $ solved for $A$

Is there a standardized way to solve $A^* B A = C $ for $A$ if $A$ is a complex and square matrice, and $B$ and $C$ are real-valued and square matrices. $A^*$ is the conjugate transpose of $A$. Is ...
ssack's user avatar
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Integral matrices commute modulo $p^k$

Given an $n \times n$ integral matrix $A \in M_n(\mathbb{Z})$, denote by $A_k \in M_n(\mathbb{Z}/p^k \mathbb{Z})$ its reduction modulo $p^k$. Now let $A, B \in M_n(\mathbb{Z})$ be be such that $A_k, ...
frafour's user avatar
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4 votes
0 answers
42 views

If $v_1 = 2v_1-3v_2 + 2v_3$ then ${v_1, v_2, v_3}$ is a linearly dependent amount.

Statement: If $v_1 = 2v_1-3v_2 + 2v_3$ then $\{v_1, v_2, v_3\}$ is a linearly dependent amount. My question: Is this statement true or not? My answer: I guess it is linearly dependent amount due to ...
J.Andreasson's user avatar
4 votes
0 answers
160 views

Methods of solving roots involving matrix determinant

I have a square matrix $F$ whose elements depend non-linearly on a complex parameter $s$. I would like to know the values of $s$ such that $\det(F)=0$, i.e., those $s$ that make $F$ singular. I would ...
George C's user avatar
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4 votes
1 answer
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A matrix problem about egienvalue and trace

Consider an $m\times m$ positive definite and Hermitian matrix $\mathbf{M}$ and an arbitrary $m\times n (m>n)$ para-unitary matrix $\mathbf{R}$, i.e., $\mathbf{R}^H\mathbf{R}=\mathbf{I}_n$. ...
LinTIna's user avatar
  • 275
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0 answers
1k views

Solving following second order matrix differential equation

Which methods (analytical or numerical) do you suggest to solve the following matrix diff. equation? And why? (Presumably analytical method) $$ \textbf{M}\ddot{V}+\textbf{C}\dot{V}+\textbf{K}V=\...
1_student's user avatar
  • 447
4 votes
0 answers
102 views

When is ${(1-t^2)^{-N/2}}{\det(f_t(A))}$ expressible as polynomial?

Given a matrix valued function $f_t:\mathbb R^{N\times N} \mapsto \mathbb R^{N\times N}$. For $f_t(A)=B$ both $A$ and $B$ are symmetric. Which properties can be assigned to $f_t$, when the following ...
draks ...'s user avatar
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4 votes
0 answers
169 views

Invertibility of infinite order matrix

how the matrix $[e^{-(x_j-x_k)^2}]$ is invertible where $\{x_j\}$ be any real sequence such that $(x_{j+1}-x_j) >0$ for all $j \in Z$ where $Z$ denotes the set of integers.
user242335's user avatar
4 votes
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70 views

Number of solutions of an equation

Fix a vector $x\in\{0,1\}^n$, and let $a$ be a random vector in $\mathbb{Z}^n_q$ for some prime $q$. Consider $y=ax$, and $S=\{x'\mid ax'=y\}$. I want to compute the probability that $\lvert S \...
Joe's user avatar
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0 answers
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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 ...
Ovi's user avatar
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4 votes
0 answers
3k views

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 ...
elmuz's user avatar
  • 41
4 votes
1 answer
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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, ...
Museful's user avatar
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3 votes
0 answers
27 views

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 ...
augustoperez's user avatar
  • 3,226
3 votes
0 answers
92 views

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 $\mathbf{J=Q^TAQ}$? ...
Firman's user avatar
  • 569
3 votes
0 answers
34 views

Move eigenvalues in triangular Rayleigh quotient while keeping Krylov relation

Suppose we have the relation $$AU_k = U_kT_k + u_{k+1}b_{k+1}^H$$ Where $U_k$ is $n\times k$ and $U_k^H U_k = \text{Id}_k$, $A$ is $n\times n$, $T_k$ is upper triangular $k\times k$, $u_{k+1}$ is $n\...
jacopoburelli's user avatar
3 votes
0 answers
75 views

A question on linear algebra (svd, etc.) and matrix optimization

Let $B$ be a $d \times d$ psd matrix and let $W$ be a $N \times d$ matrix. For any $N \times d$ matrix $A$, define $F(A) := \|B-A^\top W - W^\top A\|_F^2$ and $G(A) := \|A^\top W\|^2_F + \|W A^\top\|...
dohmatob's user avatar
  • 9,565
3 votes
0 answers
110 views

Solving an infinite system of equations

I'm hoping to solve the system of equations $$ a_1 + a_2 + a_3 + a_4 + ... = A_1$$ $$ a_1 + 2a_2 + 4a_3 + 8a_4 + ... = A_2$$ $$ a_1 + 3a_2 + 9a_3 + 27a_4 + ... = A_3$$ $$ ...$$ $$ a_1 + n^1a_2 + n^...
Caleb Briggs's user avatar
  • 1,083
3 votes
0 answers
36 views

Existence of solution(s) to a the quadratic matrix equation $P^T Q P - A^T P - P^T A + R = 0$?

While investing an optimization problem, I ended up with the following quadratic matrix equation $$ P^T Q P - A^T P - P^T A + R = 0$$ in which $Q, A$ and $R$ are $n\times n$ real matrices and $Q$ and $...
Daphne's user avatar
  • 31
3 votes
0 answers
538 views

Positive definite solution to matrix differential equation (dynamical Riccati equation)

I am wondering how I can show that the solution ($P$) to the following matrix differential equation $$\begin{align} \dot{P}(t) &= -A^{\top}P - PA + PBR^{-1}B^{\top}P - Q \\[3mm] P(t_{1}) &= S \...
SimpleProgrammer 's user avatar
3 votes
0 answers
140 views

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 ...
Florian Ente's user avatar
3 votes
0 answers
156 views

Collatz Conjecture: Understanding why for a given $K, L$, there are only a finite number of solutions

I am reading through this paper by by Simon & Weger regarding the Collatz Conjecture. I am stuck on the reasoning at the end of 2.2 Let: $n$ be a natural number. $T(n) = \begin{cases} \frac{1}{2}...
Larry Freeman's user avatar
3 votes
0 answers
61 views

How can I demonstrate this result?

From Schwinger's paper, chapter 6, this expression for the operator $U(s)=e^{-i(H_0+H_1)s}$ can be easily obtained (the commutator $[H_0, H_1]$ is unuseful, just know it's not 0). So $U(s)=U_0(s)-is\...
modellatore's user avatar
3 votes
0 answers
85 views

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 ...
maple's user avatar
  • 2,883
3 votes
0 answers
126 views

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? ...
aryan bansal's user avatar
  • 1,923
3 votes
0 answers
96 views

How many 4 by 4 real matrices $A$, up to similarity, satisfy $(A-I)(A^2 + I)^2 = 0$

So I know the minimal polynomial $m(x)$ must divide $(x-1)(x^2+1)^2$. Since we are working with $4$ by $4$ real matrices, $m(x) = (x-1)$ or $m(x) = x^2 + 1$ or $m(x) = (x^2+1)^2$ or $m(x) = (x-1)(x^2+...
jmac's user avatar
  • 151
3 votes
0 answers
342 views

Positive definiteness of solution to Sylvester / Lyapunov equations

Given square matrices $A,Q$ with $Q>0$ the matrix equation known as the Lyapunov equation is given by $$A^TX+XA+Q=0.$$ Lyapunov's theorem state that the solution $X$ is symmetric and satisfies $X&...
Conformal's user avatar
  • 1,490
3 votes
0 answers
234 views

The elements of a matrix group with order two and its centre

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 $\mathbb{F}_{7}$}\...
Mathbeginner's user avatar
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 ...
G. Trav's user avatar
  • 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 ...
user416571's user avatar
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 ...
Murad Sh-ov's user avatar
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 $$i_{0}, i_{1}, \ldots, i_{p-1} \in \{0,1,\ldots,M-1 \}, \omega = e^{-j\frac{2\pi}{M}}$$ ...
ArtificiallyIntelligent's user avatar
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 ...
Dr Krishnakumar Gopalakrishnan's user avatar
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 $3\times 9$ ...
GBaardink's user avatar
  • 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 $d$...
chausies's user avatar
  • 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\...
Bonnevie's user avatar
  • 144
3 votes
0 answers
57 views

An equation with matrices

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. ...
Shroud's user avatar
  • 1,568

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