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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Decomposition of a group element as a product of a coset and subgroup element

It is a known fact in the theory of non-linear symmetry realizations in physics that we can decompose a group element as (sum signs are implied) $$ g=e^{\theta_\alpha X^\alpha}e^{u_i t^i} $$ where $t^...
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Unique solution for A=BC with B and C unkwown

Suppose I have a system of equations $A=BC$, where $B,C$ are unknown matrices which I would like to find. For the dimension assume $A$ is $N \times L$, $B$ is $N \times K$ and $C$ is $K \times L$, ...
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Matrix notation for Stump chi^2 [closed]

For a certain minimization problem (see [$\dagger$], which I want to solve numerically with C++ matrix libraries), I have a $k\times k$ matrix $\bf{A}$ defined in terms of a $k\times N$ matrix $\bf{B}$...
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How to compute the derivative of this matrix equation

The matrix $\mathbf{A}(c)$ with the dimension $M \times N$, $c$ is a scalar variable. The matrix $\mathbf{d}$ is a constant matrix with the dimension $M \times 1$. If the formula $\frac{d\mathbf{A}(c)}...
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How to write state space equation $\dot{x} = Ax + Bu$ when additive terms are present? [closed]

Suppose there is a system of state $x$ with dynamics: $\dot{x} = Ax +Bu + C$ Where $A$ and $B$ are $n\times n$ matrices showing the dynamics of state $x$ and input $u$. $C$ is a constant $n\times 1$ ...
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How to relate spectral radius and Cauchy interlace theorem in a separate equation

Consider a a matrix $A_{n\times n}$. Also consider another matrix $M_{(n-1)\times(n-1)}$ which is obtained by deleting the $i^{th}$ row and column of $A$. I have the following equation in $M$ \begin{...
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derivative of determinant, solve singularity equation for a variable

I would appreciate any tipps on the following problem I really struggle with: For $A \in \mathbb{R}^{n_2\times n_1}, B \in \mathbb{R}^{n_3\times n_2} , \lambda \geq 0, L \geq 0.$ I want to find a ...
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What conditions are required to guarantee that my matrix is skew-Hermitian?

Consider the equation $$ \frac{\partial \boldsymbol{T}}{\partial t}=\kappa \space \frac{\partial^2 \boldsymbol{T}}{\partial x^2} $$ with $$ \boldsymbol{T}=(T_1,T_2,...T_N)^T $$ Let $$ T_i=\sum_{j=1}^...
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Second order nonhomogeneous differential equation

Let's suppose to have a system of four first order nonhomogeneous differential equations that i can regroup into a sysyem of two second order nonhomogeneous differential equations. Then I can also ...
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Reverse engineering a matrix in RREF

I know that multiple matrices can have the same RREF, but a matrix has a unique RREF, I am trying to reverse engineer a 3x3 RREF matrix to get an abstract matrix of the same size that has 2 columns ...
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Does $AX+XB=B$ have a unique solution?

Let $A, B \in \Bbb C^{n \times n}$ such that $A$ is nilpotent but $B$ is non-nilpotent satisfying $AB+BA=O$. Then, $AX+XB=B$ has unique solution. True or false? I don't understand how to approach ...
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Counterexample for a convex problem

The convex optimization problem is as follows: \begin{align} \underset{\mathbb{X},\mathbb{Y}\in\mathbb{S}_n^+}{\min}\quad &\operatorname{Tr}(X)+ \operatorname{Tr}\left(D Y \right)\nonumber\\ \...
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2 answers
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Derivative of diagonal matrix expression: $f(X)=\text{diag}(X)M^T\text{diag}^{-1}(MX)$

Let X be a vector $\mathbb{R}^{n\times1}$ and M be a constant matrix $\mathbb{R}^{n\times n}$, and given the function $f(X)$ how could i find the derivative of $f(X)$ with respect to $X$? In the ...
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Mathematical notation on a matrix

Matrix notation question (in reference to "simultaneous interconnection and damping assignment passivity based control"): We select an $n\times n$ matrix called $F_d(x)$. Let $G(x) := F_d(x)+...
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Combining two fundamental matrices

Let $\mathcal{F_{ab}}$ be the fundamental matrix obtained from images $A$ and $B$ $$ \mathcal{F_{ab}} = \begin{bmatrix} ab_{11} & ab_{12} & ab_{13} \\ ab_{21} & ab_{22} & ab_{23} \\ ...
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Some questions about the conditions for the CARE/DARE solutions

I'm trying to understand the CARE/DARE solution and have a few questions. For example, for the CARE the equation is $$A^tP + PA - PBR^{-1}B^tP+Q=0$$ where $P,Q,R,A,B$ are $n\times n$ matrices and $P,Q,...
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3 votes
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Equality of product of matrices

I have been wondering if the following statement is corrrect? Assume that $A$ is $n \times m$ matrix where $m \leq n$ and rank of $A$ is $m$, $X$ and $B$ are $m \times m$ square matrix and it is ...
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1 vote
1 answer
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Computing ABx given Ax

In general matrix multiplication does not commute, but is there any information about one ordering of the product contained in a different ordering (assuming the dimensions match)? For example, if I ...
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Existence of Solution of a matrix equation

Suppose, $Ax=b$ does not admit any solution for some $n*n$ matrix & some b in $R^n$.Does it imply $A^tx=b$ also not admit solution ? I think it may admit solution but I am unable to find out ...
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2 votes
1 answer
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How to solve for $\boldsymbol{\theta}$ in $[A]^{\theta}x = b$ if $\boldsymbol{[A]}$, $\boldsymbol{x}$, and $\boldsymbol{b}$ are known?

I am trying to find $\boldsymbol{\theta}$ when the output $\boldsymbol{b}$, input $\boldsymbol{x}$, and the matrix $\boldsymbol{[A]}$ are given. Here, $\boldsymbol{[A]}$ is a diagonal square matrix ...
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Is there a such thing as "Stacked Vector Notation"?

I was reading this link over here: https://peterroelants.github.io/posts/gaussian-process-tutorial/ . I came across the following statements: I am trying to understand how to "fill in the ...
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Relationship between a,b and c that make the equation system have infinite solutions

i have this matrix \begin{cases} \phantom{2}x_1+2x_2-3x_3=a\\ 2x_1+3x_2+3x_3=b\\ 5x_1+9x_2-6x_3=c\\ \end{cases} And the excercise says that i have to find the relationship between a, b, c that makes ...
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Solving tensor version of homogeneous Sylvester equation

I have a system that reads (in summation convention) $$ X_{j_1 j_2 ... j_N} R^{j_1}_{k_1} R^{j_2}_{k_2} ... R^{j_N}_{k_N} = X_{k_1 k_2 ... k_N} $$ where $N$ is a fixed value, $R$ a known matrix (...
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1 answer
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Proving that matrix to the power multiplied by a non-zero vector are independent from same matrices to different powers

Suppose I have a $n$x$n$ square matrix $B$ and $n$-length column vector $a \ne \underline{0}$, such that \begin{align*} B^{3}a &= \underline{0} \\ B^{2}a &\ne \underline{0} \end{align*} I know ...
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1 answer
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Weighted least squares formula

I was studying the weighted least squares algorithm and came across this formula for calculating the weighted result in terms of the original. Here $x_{LS}$ is the solution for $D=I$ I can't figure ...
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If $A$ is a square matrix

If $A$ is a square matrix, and $A=A^2$, then what would the possible values of $|A|$? I've tried to calculate it through basic mathematics, however I feel it's not appropriate... $$A=A^2$$ $$A-A^2=0$$ ...
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Finding out one combinations such that linear combinations makes zero vector.

Find a combination$~x_{1}\mathbf{\omega}_{1}+x_2\mathbf{\omega}_{2}+x_{3}\mathbf{\omega}_{3}~$that gives the zero vector with$~x_1=1:~$ $$\mathbf{\omega}_{1}= \begin{bmatrix} 1\\2\\3 \end{bmatrix}~~~~ ...
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How could I calculate the derivate an expression with a diagonal inverse function?

Given a vector $x=[x_1,x_2,...,x_n]$ and a matrix $Z$ with dimensions $n\times n$, the function $g(x)$ is described by:$\def\diag{\operatorname{diag}}$ $$ g(x)=\diag(x) \diag^{-1}(Zx)$$ Where $\diag(x)...
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1 vote
3 answers
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Locate a pinhole camera using a fiducial marker

Note: The superscript notation used refers to the frame of reference. There are three frames of reference: $w$, the world frame (in Euclidean 2-space), $c$, the camera frame (in Euclidean 2-space), ...
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Derivative of this matrix expression?

Assume that I have the following expression: $$f(x,c) = \int_{0}^{x}g(\mathbf{\Psi}(x)^\intercal\mathbf{c}) dx$$ where $f(x,c)$ is a scalar, $x$ is a scalar, $\mathbf{\Psi}(x)$ is a $N$-by-$1$ vector, ...
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Nonsingularity of sum of matrices

Consider the matrix $E$ where $E$ is defined as follows: $E=A+\gamma BC+DB-BC^{-1}DC$ where $\gamma$>0, $A,C$ are positive definite, $D$ is Hurwitz and $B$ is positive semidefinite. I have a ...
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Can we give an upper bound for the second largest eigenvalue where there is only one positive eigenvalue?

In continuation to my previous post on the nonnegative matrix I have the following follow up question: Suppose $A$ be a strictly positive matrix(entrywise) of the form $D+B$, where $D$ is a diagonal ...
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3 votes
1 answer
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Which matrix with positive entries has only one positive eigenvalue?

Let $A$ be a strictly positive matrix, by strictly positive matrix I mean that the entries of $A$ are strictly positive. Also, we assume that the entries of the matrix are from Natural numbers. Is ...
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Are there any solutions to $Ax=b$ satisfying $\Vert x \rVert=1$? [closed]

Let $$A = \begin{pmatrix} 1 & 1 & -3 \\ -3 & -2 & -2 \\ -7 & -5 & 1 \end{pmatrix}$$ be a $3$ by $3$ matrix and $$b = \begin{pmatrix} 4 \\ 6 \\ 8 \end{pmatrix}$$ be a column ...
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1 answer
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insert a matrix in-between product of vectors

Let $a=[1,2]^t$ ( $^t$ means transpose) and let $A=\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$. One can verify that $a^t a = 5$, $a^t A a = 14$, and $a^t a \cdot \det(A)= 15$. So obviously, $...
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4 votes
1 answer
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When does the following frobenius norm equality hold?

When we consider the square matrix ${\bf A}\in\mathbb{C}^{N\times N}$, then following ineqaulity always holds: $$ \left\|{\bf A^{\sf H}}{\bf A}\right\|_F = \left\|{\bf A}{\bf A^{\sf H}}\right\|_F\le\...
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Having trouble understanding the notation for matrices with seemingly incompatible dimensions in a paper; dimensions for matrix math?

Looking at equation 4 in this paper. $$ H_{aj}^{k+1} = H_{aj}^{k} \frac{\sum_{i=1}^{n}W_{ia}^{k}V_{ij}/(W^{k}H^{k})_{ij}}{\sum_{i=1}^{n}W_{ia}^{k}} $$ For more context, this is a non-negative matrix ...
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10 votes
1 answer
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If square matrices $A^2 + B^2 = 2AB$, then prove that $p_A(x) = p_B(x)$

Original problem statement: Let $A, B \in M_n(\mathbb{C})$ such that $A^2 + B^2 = 2AB$. Prove that for any $x \in \mathbb{C}$: $$det(A - xI_n) = det(B-xI_n)$$ Now the first observation, the equality ...
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Formula for matrix inverse with non-commutative entries [duplicate]

I have a square matrix $A$ with elements $A_{i,j}\in\mathbb{A}$ where $\mathbb{A}$ is a ring with with addition ($+$) and multiplication operations ($\times$). The operation $\times$ is non-...
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1 answer
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How to solve the matrix equation $\mbox{tr}(\mathbf{X}\mathbf{X}^H)\overline{\mathbf{B}}=\mbox{tr}(\mathbf{X}\mathbf{X}^H\mathbf{B})\mathbf{I}$?

I want to solve the following equation for $\mathbf{X}\in\mathbb{C}^{N\times M}$, with $M < N$: $$\mbox{tr}(\mathbf{X}\mathbf{X}^H)\overline{\mathbf{B}}=\mbox{tr}(\mathbf{X}\mathbf{X}^H\mathbf{B})\...
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1 answer
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2 identities related to determinant and exponential function

This question is from my assignment in smooth manifolds and I was unable to solve this problem. I am asking it here as I think I will not be able to solve it by myself. Question : Let $A\in M(n, \...
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1 vote
1 answer
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Equivalent parametric solution sets

I have this question that I've never seen before, I've only ever learned about how to find the parametric version of a solution set, but I've never learned how to change it in any way... I will type ...
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1 answer
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Minimal polynomial of the matrix $A = \begin{bmatrix} c & 1 & 0 & 0\\ 0 & c & 0 & 0 \\ 0 & 0 & c & 1\\ 0 & 0 & 0 & c \end{bmatrix}$

I am learning about the minimal polynomial of a matrix for the first time, but I don't understand how to find quickly the minimal polynomial of some matrices. For instance, I know that the minimal ...
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0 answers
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Calculating the unknown elements of a matrix based on a known matrix.

Suppose there is a 2$\times$2 matrix $M=\matrix[a\ ,b\ ;\ c\ ,d]$, which $$M \matrix[y_1\ ,\ y_2]^T=\matrix[y_1'\ ,\ y_2']^T$$ if the elements of the matrix M are known, is it possible to calculate ...
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1 vote
1 answer
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How to decompose matrix to its composition of rotation and scaling?

Ans: for rotation and scaling factor are $\frac{\pi}{6}$ and 8 respectively. I found a related question, but it wasn't explaining it well. I also understand scale and rotation separately but cannot ...
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0 votes
1 answer
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General term for a series of matrix multiplication

A and B are two non-zero square matrices such that ${A^2}B=BA$ and if $(AB)^{10} = {A^k}{B^{10}}$ then the value of k is? Attempt: Tried solving for lower powers of $AB$ and observing a pattern $AB=AB$...
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0 votes
1 answer
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Calculator gives unit vector, how to derive vector given only unit vectors?

My CAS HP Prime calculator gives only unit vectors when I use the function for Eigenvectors. I know how to find a unit vector, $u=\frac{v}{||v||}$, but that doesn't seem to help. For example: ...
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1 answer
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Diagonalization of $A=\begin{pmatrix}8&12&-18\\ 4&18&-20\\ 4&13&-15\end{pmatrix}$ such that $A=PDP^{-1}$, how to find P, specifically, $P_2, P_3$

Using wolfram and symbolab, diagonalization of $\begin{pmatrix}8&12&-18\\ 4&18&-20\\ 4&13&-15\end{pmatrix} \rightarrow A=PDP^{-1}\rightarrow \begin{pmatrix}8&12&-18\\ 4&...
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0 votes
2 answers
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Verifying change of basis calculation by example, my lack of understanding/mistake or Author's?

a. I got the change of coordinates from B to A, P = $\begin{pmatrix}6&0&-6\\ -1&4&0\\ 1&1&3\end{pmatrix}$ b. However, I tried to use the formula for change of basis $P^{-1}(x)=(...
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Determining value of constants in simultaneous equations using inverse matrix when determinant is zero

I've been having trouble understanding how to solve this problem: Determine the values of the real constants a and b for which there are infinitely many solutions to the simultaneous equations 2x + 3y ...
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