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Questions tagged [matrix-equations]

Questions related to equations, with matrices as coefficients and unknowns.

2
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1answer
23 views

Proving matrices equation when all the matrices in it may not be invertible

I'm reviewing linear algebra for my exams this year, and I just encountered this problem. For an arbitrary matrix, $\boldsymbol{A} \in \mathcal{R}^{m \times n}$, prove there must be a unique matrix $\...
0
votes
2answers
32 views

Determine value of '$a$' for which the system is inconsistent and has infinitely many solutions.

Consider the matrix $A$, to be equal to: \begin{bmatrix}1&2&1\\-1&4&3\\2&-2&a\end{bmatrix} Then we can rewrite this as: \begin{bmatrix}1&2&1\\0&6&4\\2&-2&...
0
votes
1answer
32 views

Gradient of $g(x) = f(Ax + b)$

I need the gradient and Hessian of the function $g(x) = f(Ax + b)$. $f:\!R^m \rightarrow \!R$, $x \in \!R^n$, $b \in \!R^m$, $A \in \!R^{mxn}$ I cannot find the expression for the ...
1
vote
2answers
13 views

The conjugate of a scalar is the same scalar in a matrix-scalar multiplication?

I am reading the book, Applied Linear Algebra and Matrix Analysis. When I was doing the exercise of Section 2.4, Page 97, I thought the equation is right as followed: $(cA)^{*} = cA^{*}$ But answer ...
2
votes
0answers
42 views

Two PDE for one matrix-valued unknown?

Let $x \in (0,L)$, $t \in (0,T)$, and let $P = P(x,t) \in \mathbb{R}^{3\times 3}$, $Q = Q(x,t) \in \mathbb{R}^{3\times 3}$, $R_0 = R_0(x) \in \mathbb{R}^{3\times 3}$ and $G= G(t) \in \mathbb{R}^{3 \...
0
votes
1answer
52 views

Solve $A = C B C^t$ for $B$

I know this question, but I would like to know the middle square matrix $B$. Given positive definite matrix $A \in \mathbb R^{2 \times 2}$ and non-zero matrix $C \in \mathbb R^{2 \times 3}$, find $...
0
votes
0answers
31 views

Partial derivative of matrix

$\renewcommand{\v}[1]{\mathrm{vec}\left(#1\right)} \renewcommand{\m}[1]{\mathbf{#1}} \renewcommand{\trace}[1]{\mathrm{trace}\left(#1\right)} \renewcommand{\diag}[1]{\mathrm{diag}\left(#1\right)}$ ...
0
votes
0answers
15 views

How many solutions to a matrix/vector multiplication?

Suppose I have a square matrix $A$ (e.g., $n\times n$) that multiplies some vector $y$ ($1\times n$) into a vector of the same arrangement $z$ ($1\times n$), such that $Ay = z$, where both $y$ and $z$ ...
0
votes
1answer
49 views

Matrix multiplication $AB=0$ so $A=0 $ or $B=0$

I have tried with the idea of $AA^T=0 $ and use the trace, but nothing changed. I have also to prove : if $AB=0 $ then $BA=0$ .
0
votes
2answers
44 views

Invertibility of $I + AB(x)$

I am dealing with a matrix $$I + AB(x)$$ where $A, B(x)$ are square $n\times n$ real matrices and $x$ is a real variable. I want to find the values of $x$ for which this matrix is singular (and then ...
1
vote
4answers
71 views

Let $A \in \mathbb R^{5 \times 5}$. Prove that $A^{4} + I \neq O$

Let $A \in \mathbb R^{5 \times 5}$. Prove that $$A^{4} + I \neq O$$ I understand that I have to start from $A^{4}+I= 0$ and come up with a contradiction. Is it the Cayley Hamilton I have to use?
3
votes
3answers
53 views

Prove or disprove: If $A$ is $n\times n$ and $\exists\;m\in \Bbb{N}:\;A^m=I_n$, then $A$ is invertible.

Is this statement true? If $A$ is an $n\times n$ matrix and $A^m=I_n$ for some $m\in \Bbb{N}$, then $A$ is invertible. My trial Let $n\in \Bbb{N}$ be fixed. Then, $$[\det(A)]^m=\det(A^m)=I_n=1.$$ ...
2
votes
2answers
91 views

Determine all $2 \times 2$ real matrices $A$ such that $(1) \ \ A^2=I$, $(2) \ \ A^2=0$

Determine all $2 \times 2$ real matrices $A$ such that $(1) \ \ A^2=I$, $(2) \ \ A^2=0$ I came across this problem recently where I have to determine all the $2\times2$ matrices satisfying the ...
0
votes
1answer
51 views

Does there exists $n\in \Bbb{N}$ such that $A^n=I_2,$ where $A=\begin{bmatrix} 0 & 0\\ 3 & 1 \end{bmatrix}?$

Here's the matrix $$A= \begin{bmatrix} 0 & 0\\ 3 & 1 \end{bmatrix},$$ where $I_2$ represents the $2\times 2$ identity matrix. My trial I'm claiming that there do not exists $n\in \Bbb{N}$ ...
0
votes
0answers
12 views

Matrix Derivative of Fisher Discriminant Analysis

Let $Z_c \in \mathbb{R}^{D\times N_c}$ is column matrix which includes mapped data of class C, and $\alpha_c = \frac{1}{N_c}$ where $D:$ dimension, $N_c:$ data number of class C. $\mathbf{i}:$ One ...
3
votes
0answers
41 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 $\mathbb{F}_{7}$}\...
0
votes
1answer
61 views
+100

Is there a fast way to compute the lowest eigenvalue of this symmetric PD matrix in this specific scenario?

Consider $$C = A^H D A + M$$ where $A$ is a $m \times m$ unitary matrix. $D$ is a $m \times m$ diagonal matrix with entries either $0$ or $1$. The number of $1$'s is $n \ll m$. $M$ is a $m \times ...
0
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0answers
6 views

Specific Sylvester equation. Existence without uniqueness

I have been looking at this specific case of a Sylvester equation for the square matrix $X$, $$ AX-XA=-A, $$ given a nilpotent square matrix $A$. For a general Sylvester equation $$ AX + XB = C, $$ ...
0
votes
1answer
29 views

Solve symbolic Sylvester-like equation in MATLAB or MAPLE

I'm looking for a way to solve a symbolic Sylvester-like equation in MATLAB or MAPLE (or any other available tool). In particular, I have the following equation, $$AX+XA=B$$ where, $A$ has some ...
0
votes
1answer
13 views

For what value of k is the system of equations consistent?

For $$x − y + 2z = −2$$ $$2x + 3y + 4z = 7$$ $$4x − 7y + 5z = k$$ $$8x − 4y + 6z = 2$$ Using Gaussian Elimination, I first get the below by applying $R2 - 2R1$, $R3 - 4R1$, $R4 - 8R1$ \begin{...
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votes
0answers
14 views

Solving a linear equation using columns and rows of matrix

I have a system of equations that I am trying to approx. an answer to. I don't do any high level mathematics, so excuse any formatting errors. Find every team's individual match score. vars = $...
0
votes
1answer
22 views

Linear transformation with range of a vector [duplicate]

enter image description here What are the steps or any guidance in solving this problem? I'm very confused
4
votes
3answers
130 views

How to solve $A^{\frac 12} B A^{\frac 12} = C$ for $A$?

Suppose that matrices $A,B,C$ are symmetric and positive definite. Then, $A$ has a unique, positive square root, which we call $A^{\frac 12}$. If $$A^{\frac 12} B A^{\frac 12} = C$$ then can we write ...
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votes
0answers
20 views

$AX=ZB$ Linear equation solution

If $A$, $X$, $Z$ and $B$ are $4 \times 4$ homogeneous matrices (transformation matrix). Looks like this: \begin{bmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{...
0
votes
1answer
43 views

Two matrices whose product is equal to the identity matrix

I need to multiply two matrices one we call P type 1x4 another called Q type 4x1 I cannot find values that will give me an identity matrix as a result when I multiply PQ together. Where zeros are ...
0
votes
1answer
51 views

Solving an undetermined or overdetermined system of equations with constraints

I have a table that looks like this: I would like to determine the values for each of the different categories in the columns, such that col1*col2*col3 equal what'...
2
votes
0answers
43 views

Conditions for solving generalized Sylvester matrix equation XA + BX + CXD = E

In relation with an observation problem I have the matrix equation (1) $XA + BX + CXD = E$ where all the matrices $A$, $B$, $C$, $D$, $E$ can be assumed real, square and known, whereas $X$ is the ...
-3
votes
2answers
33 views

Is it possible to find two matrices

Is it possible to find matrices $M$ and $N$ such that $MN=O$ and $NM=I$ where $0=$ the $3\times3$ zero matrix and $I=$ the $2\times 2$ Identity matrix
1
vote
2answers
36 views

Matrices commuting with the matrix of ones

Let $J_n$ be $n \times n$ matrix of ones, so that $J_{ij} = 1$ for all $i,j \in\{1,\ldots,n\}$. I am interested to find the class of matrices which commute with $J_n$, i.e. $$M J_n = J_n M.$$It is not ...
0
votes
3answers
23 views

Row reduction as matrix multiplication help?

Let A = \begin{bmatrix}1&2\\-1&1\end{bmatrix} Give an elementary matrix M such that MA= \begin{bmatrix}1&2\\1&5\end{bmatrix} I'm trying to figure out what the values for M would be, ...
0
votes
1answer
28 views

Eigenvectors reversed question

I was given this $$M=\begin{pmatrix}-1 & x\\ -5 & y\end{pmatrix}$$ which has eigenvectors $$V=\begin{pmatrix}1\\ 1 \end{pmatrix}$$ and $$W=\begin{pmatrix}5\\ 1 \end{pmatrix}$$ I'm supposed ...
3
votes
1answer
17 views

How to show $A=\begin{bmatrix}A_1\\A_2\end{bmatrix}$ is non-singular when $N(A_1)=R(A_1^T)$?

Suppose $A=\begin{bmatrix}A_1\\A_2\end{bmatrix}$ is a block matrix such that $N(A_1)=R(A_2^T)$. How can we show it is nonsingular. My try: We know that a matrix is nonsingular if $Ax=0$ has only the ...
0
votes
0answers
24 views

Thomas method for Crank–Nicolson scheme with a central finite analog. How to get the boundary values of the coefficients?

The crank nicolson scheme: $$\begin{cases}\Large \frac{w_i^k - w_i^{k-1}}{h_{t}} = \frac{m}{2c} (\frac{w_{i+1}^k - 2w_i^k + w_{i-1}^k}{h_x^2} + \frac{w_{i+1}^{k-1} - 2w_i^{k-1} + w_{i-1}^{k-1}}{h_x^2}...
1
vote
1answer
50 views

Factoring a matrix as the product of block triangular and diagonal matrices.

How can I check that the matrix $$K = \left[\begin{array}{c|cc} 1 & 0^{\mathrm T}_m & m \mathbf u^{\mathrm T} \\ \hline 0_m & I_m & I_m \\ m \mathbf u & I_m & O_m \end{array}...
0
votes
1answer
51 views

Writing a matrix in an alternative form with a Kronecker product.

I need to express the matrix \begin{equation} \begin{bmatrix} I & A \\ A^T & O \\ \end{bmatrix} \end{equation} where $$A = \begin{bmatrix} m\textbf{u}^T\\ I_m\\ \...
0
votes
1answer
39 views

Inverting a matrix with the same diagonal entries in a particular form

Hi I'm struggling with this inversion and any help would be greatly appreciated. I want to invert the following $\mathbb R^{m\times m}$ matrix \begin{bmatrix} 1 + m & m & \dots &...
0
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0answers
31 views

Solve simple nonlinear equations in the form [A]x=b

I have a simple set of nonlinear equations 1) 3x = 30 2) x+2y = 20 3) x + y*z = 15 Clearly the solution to this is (10,5,1) but I want to find a robust way to solve this type of problem [A]x=b (...
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vote
0answers
27 views

Solving a simple system of equations through matrix operations

I'm having trouble with some basic linear algebra. My best searching doesn't exactly come up with the solution I'm looking for. My problem is as follows. $$ \begin{bmatrix} y_1 \\ y_2 \\ \end{...
1
vote
1answer
37 views

How to find the reverse of a formula when dealing with matrices

I currently have a formula that deals with animation. I'm not even quite sure what I'm asking but I'd rather just show what the issue is by sharing this google sheet with formula. I am dealing with ...
0
votes
0answers
15 views

Convergent series description of ratio of two bilinear forms

I have to numerically calculate the ratio of two bilinear forms: $\frac{x_1}{x_2} = \frac{v^T U_1 v}{v^T U_2 v}$, where $U_1$ and $U_2$ are unitary matrices. Both bilinear forms $x_1$ and $x_2$ are ...
0
votes
0answers
15 views

Finding $Y$ which $D^{-1}E = P$ and $D_{ii} = \sum_j E_{ij}$ and $E = exp(Y^TY)$.

I'm seeking for a matrix $Y$ which when I calculate the matrix $exp(Y^TY)$ and then normalize it with every row, it would be equal to a known matrix $P$ ($P$ is also row normalized and that's why I ...
0
votes
0answers
17 views

Question about an equationsystem of marices

My problem is the exact same as this one: Solving simultaneous equations of matrices In the solution the following happens $$-(BX+B\cdot BY=B\cdot B+2EB)$$ $$-->$$ $$Y-B\cdot BY=B-B\cdot B -2EB$$...
2
votes
1answer
43 views

Solving the equation $ XA = A + 2X $

How to solve $XA = A + 2X$ if $A$ being: \begin{equation} A= \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & 1 \\ 0 & 1 & 1 \\ \end{bmatrix} \\ \end{equation} ...
1
vote
0answers
27 views

Noncentral Wishart Expected Value - Solving a Matrix Integral

Let $\mathbf{V} \sim ncWish\left(\nu_1, \, \mathbf{\Sigma}, \, \frac{\nu_1}{\nu_2-p-1} \mathbf{\Theta}\right)$ follow a noncentral Wishart distribution according to Theorem 3.5.1. in Gupta, Nagar - ...
3
votes
3answers
101 views

For matrices, under what conditions can we write $AB = BC$?

If $A$ and $B$ are real matrices, when does there exist a matrix $C$ such that $AB = BC$? I understand that there is the case where $A$ and $B$ commute so $AB =BA$, but is there a more general rule (...
1
vote
1answer
28 views

$3 \times 3$ Diagonal matrix as the square of an $3 \times 3$ non diagonal matrix

It is easy to see if the matrix is $2 \times 2$, by just considering a system of equation with $4$ equations. However, for a $3 \times 3$ matrices, a system of equation with $9$ equations might not ...
0
votes
0answers
34 views

Least Squares Normal Equations in Explicit Form

I am struggling with the following least squares problem: Find the minimiser x* $\in \mathbb{R}^{m}$ of $$F(\textbf{x})=||\textbf{b}-A\textbf{x}||_2$$ where $A \in \mathbb{R}^{(m+1) \times m}$, $...
0
votes
0answers
18 views

How to get the partial information matrix from the covariance matrix

I have four random variables${\left( {x,y,z,\varphi} \right)^T}$. And I know its covariance matrix $Cov=\left[ {\begin{array}{*{20}{c}} {\sigma _{xx}^2}&{\sigma _{yx}^2}&{\sigma _{zx}^2}&{...
2
votes
1answer
32 views

About a spectral norm estimation

Consider a vector $x \in \mathbb{R}^p$ and say we have $k$ matrices $A_i \in \mathbb{R}^{p \times n}$. Now consider a matrix $Y := \Big [ x^\top A_i \Big ]_{i=1}^k$ whereby we indicate that $Y \in \...
0
votes
1answer
24 views

How to find all fixed points for this problem?

Find all fixed points of the below function $$f:X\rightarrow X$$. $$X=R^N$$ and $$d(x,y)\equiv\Vert x-y\Vert_2$$ and $$f(x)\equiv Ax$$ where $$A\equiv \begin{bmatrix} 1 & 1 \\ 0 &1 \\ ...