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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Find optimal Vector for a dataset equation problem

Is there an algorithm or efficient efficient way to solve the following equation $S= \frac{ \begin{bmatrix} R_{1} & . & . & R_{m}\\ \end{bmatrix} \begin{bmatrix} P_{1}\\ . \\ . \\ P_{n}\\ ...
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1answer
45 views

What does it mean if a vector name has an → above its name

I am doing a course and one of the mathematical algorithms which were mentioned included the name of a column vector (a m*1 Matrix) with an arrow (→) above its name. What does it mean?
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Does a reduced row echolon only have one solution?

In general, when finding the reduced row echolon for a matrix, is there only one solution, even if there is no solution? $$\begin{pmatrix}2&3&5\\ \:-2&-3&-3\end{pmatrix}$$ I got the ...
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2answers
38 views

Linear Algebra Matrices to equations

Show that $$\operatorname{det}\;\begin{bmatrix} 1&1&1\\x^2&y^2&z^2\\x^4&y^4&z^4 \end{bmatrix}=(y^2-x^2)(z^2-x^2)(z^2-y^2)$$ I am doing my homework,and this question came up. ...
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1answer
55 views

Minimize $\mathrm{tr}(B'XB)$ where $X$ is solution for DARE

For a given $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times m}$, where $(A,B)$ is controllable $$\begin{array}{ll} \underset{X \in \mathbb{R}^{n\times n}}{\text{minimize}} & \...
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Propagating in time a Nonlinear Dynamically Inverted (NDI) System

Background Suppose I have a nonlinear system given by $\dot{x}=f(x)+G(x)u$ $y=Hx$ where $x$ is the state, $y$ is the output, $G$ is a control matrix. This form is identical to how one would ...
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1answer
48 views

Section to Skew-Symmetrization Map

Let $A$ be an $n\times n$ matrix skew-symmetric matrix. Define the map $\mathbb{R}^{d^2}\to Skew_d$ by $$ B\mapsto B^{\top} - B. $$ Does this map have a continuous right inverse?
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If I have a matrix in RREF is there a quick way of pulling out vectors which are not in the span of the row vectors of A?

Say I am looking for the kernel of a linear function $\mathcal{l}_A$: $\mathbb{R}^n $->$\mathbb{R}^m$ given by the $n \times m$ matrix A. Then say A is row-equivalent to some matrix B in RREF. Is ...
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an upper bound for summation with matrix

For $i=1,\dots, m$, $f_i(x): K\subseteq \mathbb R^n \to K := A_ix+b_i$, where for all $i$, $A_i$ is a matrix of order $n\times n$, whose all eigenvalues are strictly inside unit circle. Could anyone ...
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1answer
28 views

3d matrix multiplied by a 2d matrix

Edit: There are one 3-d matrix $A = [ A_1 \ A_2 \ ... \ A_K]\in\mathbb{R}^{K\times D\times K}$ and $A_i\in\mathbb{R}^{D\times K},i=1,...,K$, and a 2d matrix $B = [B_1 \ B_2 \ ... \ B_k ]\in\mathbb{R}...
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1answer
17 views

Why does the inverse of this mapping from a square to a trapezoid not exist?

I am trying to compute an inverse mapping. I shall start with the forward mapping. Consider the parametric coordinates $(r,s)$: $r \in \mathbb{R}\wedge[-1,1]$ $s \in \mathbb{R}\wedge[-1,1]$ Then ...
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1answer
27 views

What are the meanings of matrix constructs of pre-multiplication of the matrix transpose and post-multiplication by the matrix itself.

In many books and articles I sometimes see similar matrix constructs, multiplication of the matrix transpose (or inverse), some other matrix, and then the first matrix itself, like this: $M^T AM$ or $...
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1answer
31 views

Matrix non-singular proof

I have one question of how to derive the nonsingularity of one matrix. Here's the matrix I'm interested in: \begin{align} A = I + SHFG, \end{align} where $A \in \mathcal{R}^{m \times m}$, $I\in \...
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Jacobian and Inverse Jacobian

I am new to Jacobians and still trying to understand how they work. My understanding so far is as follows: Suppose I have a function $f$ expressed as: $$f(b_1(a_1,a_2),b_2(a_1,a_2),b_3(a_1,a_2))$$ ...
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How to utilize the special feature of this recursive problem to reduce computational complexity?

Assume $A$ is a $n \times n$ matrix of non-negative numbers. $A_i$ is the $i$-th row of $A$. $(a_1, \ldots, a_n)$ and $(b_1, \ldots, b_n)$ belong to $\mathbb R_+^n$. $X= [x_1, \ldots, x_n]^\intercal$ ...
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Assume $AX = C$. How to determine which entry of $BX - D$ is non-negative?

I posted this question on https://scicomp.stackexchange.com, but seems to receive no attention. As long as I get answer in one of them, I will inform in the other. Let $A,B$ be $n \times n$ matrices ...
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28 views

Matrix positive definite lemma

Suppose I have a matrix of size $(n \times p) $, denoted $B$, and a symmetric matrix of size $(n \times n) $, denoted $A$ with $n>p$. If I know that the matrix $BB'A$ is positive definite, can I ...
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1answer
34 views

Derivative of $\text{Tr}[B X^T A X^{-1}]$

Let $A, B, X \in \mathbb{R}^{n \times n}$ and assume that $X^{-1}$ exists. Derive $\frac{\partial K}{\partial X}$ where $K(X)= \text{Tr}[B X^T A X^{-1}]$ I have tried the following so far ($U = B X^...
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Derivative of Product of Matrix with Moore-Penrose Inverse

Assume that $A(x) \in M_{m,n}$ is given and depends on some $x \in \mathbb{R}$. Let $A^\dagger(x) = [A(x)^\star A(x)]^{-1} A^\star(x)$ be the Moore-Penrose pseudoinverse of $A(x)$. Define $G(x) = A(x)...
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1answer
58 views

Matrix notation for scientific papers

I am writing a research paper for an engineering journal and I am having difficulty writing a couple of simple matrix equations. For equation 1, I have a $4 \times 4$ transformation matrix $T$. I ...
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Column space of matrix

Prove that if $A$ is an $m \times n$ matrix over the field $\mathbb R$ of real numbers, then $\mathbb R^n = \operatorname{Col}(A^T) \oplus N(A),$ where $\operatorname{Col}(A^T)$ denotes the column ...
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1answer
30 views

Matrix is computed are permuted

what happens to the matrix of a linear transformation on a finite dimensional vector space when the elements of the basis with respect to which the matrix is computed are permuted among themselves?...
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26 views

Solving a linear matrix equation with both left and right multiplication of unknown

I would like to solve a matrix equation of the form $$ \mathbf{A} \mathbf{X} + \mathbf{X} \mathbf{A}^T = \mathbf{B} $$ where $\mathbf{A}$ and $\mathbf{B}$ are known $n \times n$ matrices, and $\...
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1answer
48 views

Deriving a matrix inequality related to stability theory

I am reading a paper related to control theory and struggle to understand a matrix inequality derivation that is briefly introduced by the author: Given that we have the following equations: $ x(k+1)=...
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1answer
10 views

Pseudoinverse of block diagonal matrix

Suppose I have some block diagonal matrix $A$, defined as: $A = \begin{bmatrix} A_1 & 0 & ... & 0 \\ 0 & A_2 & ... & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 &...
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3answers
58 views

Finding matrix A with double eigenvalue -3 and A$\begin{bmatrix}1 \\ 2 \\ -2\end{bmatrix}$ = $\begin{bmatrix}6 \\ 12 \\-12\end{bmatrix}$

A is a symmetric matrix size of 3x3 that has two eigen values of -3 and we have the equation A$\begin{bmatrix}1 \\ 2 \\ -2\end{bmatrix}$ = $\begin{bmatrix}6 \\ 12 \\-12\end{bmatrix}$. The problem ...
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35 views

Solving a matrix equation given an eigenvector

If $v(z=0)$ is chosen to be an eigenvector of $A$ describe the solution evaluated for $v(nδ)$, for any integer $n$ for the following: $$(NI-A)v(z+δ)=(NI+A)v(z)$$ where $N$ is an arbitrary number. ...
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1answer
18 views

How to discuss the solution of this equation system in Function of Lambda

How I should discuss the solution of the following system of equations depending on the parameter of Lambda $x-y+z=\lambda \ , \ \ \lambda \in \mathbb{R} \\ 2x-3y+4z=0\\ 3x-4y+5z=1$ The only way ...
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1answer
41 views

Eigenvectors of Matrix A [closed]

For a matrix Q prove the eigenvectors of kI-Q are equal to that of Q
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Relation between Leslie dinamical system matrix and homogeneous 3rd degree recurrence relation

For a simple Leslie Matrix and asociated system we can get the value of a vector $x_0$ by using $X_{n+1}=AX_N$. For example with the following system $X_{N+1}=AX_N=\begin{pmatrix} 0 & 3 & 5\\ ...
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15 views

Find rigid body transform $T_b$, given transforms $T_a, T_c$, where $T_c = T_b^{-1}T_aT_b$ for $T_b$. i.e. find change of basis.

Let $\mathbf{T}_a, \mathbf{T}_b, \mathbf{T}_c \in SE(3)$ be rigid body transforms, and; $$ \begin{equation} \mathbf{T}_c = \mathbf{T}_b^{-1}\mathbf{T}_a\mathbf{T}_b. \label{eq_basis} \end{equation} $...
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1answer
46 views

Solving matrix quadratic equation [closed]

Let $\mathbf{A}, \mathbf{C}\in \mathbb{R}^{n\times n}$. Given a skew-symmetric matrix $\mathbf{G}$ I am looking for any numerical procedure so solve the following quadratic equation. $$ \mathbf{B}^\...
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2answers
24 views

Does a square matrix satisfy only its characteristic equation?

Say that we have a square matrix of order $N \times N$ and is NOT a diagonal matrix. Now can this matrix satisfy a polynomial of degree $N$ other than its own characteristic equation?
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56 views

Comparison of two least-squares optimization problems

I have come across two least square minimization problems. The first one is: $$\min_{\beta\in \mathbb{R}} \lvert y_j-x_j\beta\rvert, \quad \text{where}\ j = 1, \dots, n.$$ Here $y$ is the dependent ...
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2answers
59 views

What does $[T]_{\cal BB}$ mean?

I have been given a matrix P where the columns represent a basis in B. I have also been given a matrix A which is the standard matrix for T. I am then supposed to calculate the matrix for T relative ...
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1answer
58 views

How to write the first term of Taylor expansion for a function matrix?

Assume $F(A)=\left\| A^N v-w \right\|^2$ where $A$ is a square matrix and $v$ and $w$ are two constant vectors and $N$ is a positive integer. Also, the norm is the Euclidean norm. How to write the ...
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least squares: solve for b hat without knowing y

I am trying to solve for $\hat{B}$ without knowing the $y$ vector. I know $\hat{B}=((X'X)^{-1})X'y$ and i am given the norm of $y$, the projection of $y (X\hat{B})$, and the $X$ matrix (non-invertible)...
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53 views

Show that $p^T \Sigma^{-1} p \leq k^2$ is equivalent to $w^Tpp^Tw \leq k^2 w^T \Sigma w$

I am trying to understand a research paper 1 (Eq 11 and Eq 10) which states the solution to the following 2 optimization problems are equivalent. $$\begin{array}{ll} & \min_{p} p^Tw \\ & \...
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1answer
27 views

If there exists $E$ and $F$ square matrices such that $EA=B$ and $FB=A$, then there exists invertible matrix $G$ such that $GA=B$?

there exists $E$ and $F$ square matrices such that $EA=B$ and $FB=A$, then there exists invertible matrix $G$ such that $GA=B$? I'm needing a way too detailed proof or a simple counter-example, 'cos ...
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19 views

What is Wrong with the Solution to this Discrete Dynamical System?

I am supposed to solve the discrete dynamical system 𝑥_k+1=A*x_k, and I have been given both the matrix A and x_O (please see the picture below). I started out by calculating the eigenvalues and then ...
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31 views

3d Camera View Volume

Suppose I have 80x80x80 cube with the center point located at (0,0,0) as the condition drawn below: : I want to convert this parallel projection view volume into a canonical view volume. I have ...
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1answer
30 views

Expressing rows in terms of other rows in a matrix

how do I explicitly prove that a matrix of this kind: $$A=\begin{pmatrix}1&1&0&0&0\\ \:1&1&0&0&0\\ \:0&0&0&0&0\\ \:0&0&0&0&0\\ \:0&...
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17 views

Help deriving basic equations for matrix tri-factorization using gradient descent

So for some background, I only have some basic understanding of matrix calculation. Based on what I remembered from backpropagation, I successfully derived equations for Matrix factorization using ...
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1answer
34 views

Taylor-formular for Matrices

Let $A\in\mathbb{C}^{n\times n}$. What does it mean to make a Taylor expansion of $A$ around a point? I assume that the induced linear map is what should be expanded, but I have never seen a Taylor ...
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2answers
60 views

Solving a linear matrix ODE

I have an equation of the form $$\dot{P}(t) = A^T P(t) + P(t) A + Q$$ I also have an initial condition $P(0)=P_0$ Only the articles on the algebraic Riccati's equation mention equations of this ...
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1answer
24 views

Word for matrices with following property

Suppose I have two matrices $A$ and $B$ and they satisfy the property that: $A$ has non-zero values on row $j$ $\implies$ Row $j$ of $B$ contains only zero values. $B$ has non-zero values on row $j$ ...
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0answers
10 views

Numerically finding G when GQG^T = Z

I have to find $\Gamma$ for this equation: $$ \Gamma_k Q_k \Gamma_k^T= \int_{t_k}^{t_{k+1}} e^{F(t_{k+1} - \sigma)} G(\sigma) Q_k G(\sigma)^T (e^{F(t_{k+1}-\sigma)})^T d\sigma$$ Here Q is a ...
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1answer
26 views

Eigenvalues of a square matrix are the roots of a polynomial.

Let $p(t)$ be a polynomial, $A$ be an $n\times n$ matrix such that $p(A) = 0$. Then the eigenvalues of $A$ are all roots of $p(t)=0$, i.e., $p(\lambda_i) = 0$ for each eigenvalue of $A$. I know I ...
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4answers
77 views

With a given matrix $A \in M_3(\mathbb{R})$ show that $A^{2009} + A^{2008} = 2 ^{2008} (A + I_3)$.

I am given the matrix: $$A = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ \end{pmatrix}$$ I have to show that the following is true: $$A ^ {2009} + A ^ {2008} = 2 ^...
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1answer
21 views

Finding the decision boundary function for a very basic neural network

I have a simple neural network and want to draw its decision boundary. 2 input neurons(x,y), 3 hidden neurons, and 2 output neurons. So essentially drawing a line for outputNeuron1 - outputNeuron2 = ...

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