Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [matrix-equations]

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

0
votes
2answers
35 views

Matrix equations, simplify them.

I have some equations and I don't know am I doing the simplification right. Can someone check it? For example, we have an equation $AX = 4X + B$, where $A$ and $B$ are matrices. So, what I have done ...
0
votes
0answers
16 views

is this the correct way to solve this equation, to find $W^*$

$$\min_{W} ||XW-X||_F^2+p_1||W||_1+p_2R(W), W\geq0$$ Find derivative of equation above equal to $0$ $$2X^T(XW^*-X)+p_1||W||_1+p_2R(W)=0$$ $$2X^TXW^*-2X^TX+p_1||W||_1+p_2R(W)=0$$ $$2X^TXW^*=2X^TX-p_1|...
0
votes
0answers
21 views

Need help to find derivative of matrix norm

$$\min_{W} ||XW-X||_F^2+p_1||W||_1+p_2R(W), W>=0$$ Guys i need help how to find first derivative of this equation $$R(W)=Tr(W^TX^TLXW)$$ L=Laplacian matrix
0
votes
0answers
23 views

How to get W* form minimize function

I need help to solve this equation $$\min_{W} ||XW-X||_F^2+p_1||W||_1+p_2R(W), W>=0$$ X size dxn (d = number of feature & n = number of data) Example input: X is random matrix 3x5 $$R(W)=Tr(W^...
2
votes
4answers
71 views

Number of solutions $X$ to $AX=XB$ in $\mathbb F_2$

It is a well-known theorem that in an arbitrary field $F$, if $A$ is an $m\times m$ square matrix and $B$ is an $n\times n$ square matrix, then there is a unique $m\times n$ solution $X$ to the ...
1
vote
1answer
35 views

Regarding a real $n \times n$ matrix $A$ satisfying $A^6 = -A^2$

The Problem: Suppose $A$ represents a real $n \times n$ matrix satisfying $A^6 = -A^2$. (a) Prove that if $A$ is symmetric, then $A = 0$. (b) Prove that if $n$ is odd, then $A$ is not invertible. (...
0
votes
0answers
27 views

Intersection of parameterized cosets in a free-abelian group

Let $L_1,L_2$ be subgroups of $\mathbb{Z}^m$, and let $\mathbf{P}_1,\mathbf{P}_2$ be $r\times m$ integer matrices. Then, it is straightforward to check that the set \begin{equation} S_{\{1,2\}} := \{...
0
votes
2answers
25 views

finding Ker(T) of a parameter's linear transformation

I am suppose to find the ker(T) of linear transformation of: $$ G\begin{pmatrix}a & d \\ c & b\end{pmatrix}= a+\frac{b+c}{2}x+\frac{b-c}{2}x^2 $$ the form $T:V \to W$ My problem is that I ...
2
votes
2answers
27 views

How to rewrite matrix formula for Diagonalizable matrix $A=PDP^{-1}$

I am working on an old exam containing a question about Diagonalizable matrix, I am quite confident about the subject overall but there is one simple thing that bothers me, a lot! We are given the ...
-1
votes
0answers
25 views

If you compress a matrix does some math proprietes remain the same? [closed]

Let's say you have a 8x8 matrix. If you compress it in a 4x4 matrix by merging every 4 elements in it. The question is, does something remain the same? for example the determinant of the matrix or ...
0
votes
0answers
19 views

Alternatives in Farka's Lemma as boudaries

I am attempting to solve a problem in the field of Economics, and for that purpose I have devised the following lemmas. Lemma 1: Let $A(x)$ be a square matrix and let $x^*$ be such that $A(x^* )z=0$ ...
2
votes
2answers
21 views

Matrix quadratic form expansion question

I'm trying to do a question and within it, I need to expand a matrix quadratic form: $\frac{1}{2}(\vec{y} - \vec{x})^{T} \Sigma (\vec{y} - \vec{x})$ In my working out, I think that the following is ...
-1
votes
1answer
41 views

Prediction methods to resolve differential equation.

How resolved this difference equation used prediction methods? $$X' = \left[\begin{array}{ccc}1&-1&-2\\1&3&2\\1&-1&2\end{array}\right]X + \left[\begin{array}{c}t^{2}\\t+2\\2\...
0
votes
0answers
32 views

Solving a matrix equation with overdetermined case

I have a matrix equation $\begin{bmatrix} a& e& i\\ b & f & j\\ c & g &k\\ d & h& l\end{bmatrix}.B=\begin{bmatrix} d& c&i\...
0
votes
0answers
16 views

Given a mathematical formula or definition, how to find how is it called and where to find the mathematical theory that covers it?

I am trying to solve a certain physical problem. I found a way to express the problem as one of solving the following system of equations: $$\begin{pmatrix}A_{s1}\\ A_{s2} \end{pmatrix}=\begin{pmatrix}...
0
votes
3answers
40 views

Find $\lambda$ and solve the matrix

Find $\lambda$ and solve the matrix. I have 4 equations: \begin{cases} x+y+z-t=2\\ x+y-z+t=2\\ 3x+y+z+t=\lambda\\ x-y+z+t=2\\ \end{cases} I've got $$t=\frac{\lambda -6}{4}; \ z=\frac{6-\lambda}{4};\ ...
1
vote
1answer
23 views

Names for couple of structured matrices

In my work I have come about two type of matrices: 1-Let $U_{np}\in \mathbb{R}^{n\times p}$ be such that $(U_{np})_{ij}=p-j+1$, e.g. $U_{53}=\begin{bmatrix}3 & 2 & 1\\ 3 & 2 & 1\\ 3 &...
1
vote
1answer
25 views

A commutativity quirk in a matrix “differential equation”

Let $F:\mathcal{B}\to\mathbb{C^{n\times n}}$, where $\mathcal{B}=\{z\in\mathbb{C}:|z|<R\}$ be given by $F(z)=\sum_{k\geq 0} A_k z^k$, with some radius of convergence $R$. (edit: $A$ is obviously ...
2
votes
1answer
45 views

Let $A, B$ be $n\times n(n\ge 2)$ nonsingular matrices with real entries such that $A^{-1} + B^{-1} =(A+B)^{-1}$

Whole question looks like-Let $A, B$ be $n\times n(n\ge 2)$ nonsingular matrices with real entries such that $A^{-1} + B^{-1 }=(A+B)^{-1}$, then prove that $\operatorname{det}(A)=\operatorname{det}(B)$...
0
votes
1answer
38 views

Give the linear systems: $(i) x_1+2x_2=2,3x_1+7x_2=8\\(ii) x_1+2x_2=1,3x_1+7x_2=7$ [closed]

Give the linear systems: (i) $x_1+2x_2=2\\3x_1+7x_2=8$ (ii) $x_1+2x_2=1\\3x_1+7x_2=7$ Solve both systems by incorporating the right-hand sides into a $2\times2$ matrix $B$ and computing the ...
0
votes
1answer
42 views

Calculate the product $A^{10}v$ where $A$ is a $2 \times 2$ matrix and $v$ is a vector $[4 \; 4]^T$ [duplicate]

Following on from the title, can someone suggest how to proceed with this one. $$A=\begin{bmatrix}1&1\\4&1\end{bmatrix}$$ and $$v = [4 \; 4]^T?$$
0
votes
0answers
18 views

Converting Cube Coordinates of Hexagon Maps into Matrices

I came across an article here that has discussed three types of hexagonal coordinate system (square, parity, and cube) as shown in this figure and their transformed matrices (equation (1), (2), and (3)...
0
votes
0answers
14 views

rearranging a scattering matrix

Please I need some assistance. I am formulating a scattering matrix problem for some multilayers. In my formulation, I ended up with the following matrix equation $$ \begin{bmatrix} c^{ ′+}_2 \\ ...
0
votes
2answers
36 views

Is $r.b*=b*A$ true?

Let $a, b \in \Bbb C^n$ and $A\in \Bbb C^{n \times n}$. If $b^*\cdot a=1$ and $r=b^*\cdot A\cdot a$, is it true that: $r\cdot b^*=b^*A$?
1
vote
0answers
26 views

When a linear system has one solution

I have a matrix of coefficients of a linear system. How can I prove that the system admits exactly one solution whatever the known terms of the system? Is it sufficient to prove that the matrix is ...
0
votes
0answers
22 views

How do I optimize Ax = b such that every element of b is bigger than 0.5 and x has the lowest possible norm?

As the title says I want to find x such that b, given by $Ax = b$ has ideally every element above 0.5 And x has the lowest possible norm. A is a symmetric definite positive matrix Thanks!!
1
vote
0answers
50 views

Is there a way to avoid computing the inverse in the problem $\mathbf{y}=AB^{-1}\mathbf{x}$?

I'm writing some code in Python that has to deal with very large matrices so it is virtually impossible (atleast very sub-optimal) to compute inverses of matrices. In an earlier problem I ran into a ...
0
votes
1answer
22 views

Change of coordinate transformation matrix

I want to convert the following matrix: $$ \left[ \begin{matrix} 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1\\ ...
-1
votes
1answer
55 views

How to calculate sparseness and sparse density of the matrix having elements 'Zero', 'One' and 'X'.

I would like to calculate number of elements in a matrix that are equal to Zero or One. The goal is calculate density of elements in a matrix that are equal to zero or one. I use this to calculate ...
2
votes
1answer
34 views

MatLab: Not Enough Eigenvectors for (Repeated) Eigenvalues

I am trying to make a code that matches the eigenvalues to eigenvectors for one of my projects and I am new to MatLab. I get a 3x3 matrix output when it comes to eigenvalues and a 3x2 when it comes to ...
1
vote
1answer
53 views

Matrices of different dimensions in a matrix equation

Given two matrices A, B, respectively, and an equation XA=B; how can we obtain X and discuss the system? \begin{pmatrix} 1 & a\\ a & 1 \\ \end{pmatrix} \begin{pmatrix} ...
7
votes
1answer
102 views

Solving the matrix equation $X^tA+A^tX=0$ for $X$ in terms of $A$

Suppose that I know $A$. And all matrices in the equation are square matrices. I want to solve for $X$ given that $$X^tA + A^tX = 0$$ I'm not really good at matrix calculus. Is it possible to solve ...
0
votes
1answer
44 views

Find diagonal matrix $D$ such that $A D$ is Hurwitz

Let $A \in \mathbb{R}^{m \times m}$. Give necessary and/or sufficient conditions for the existence of a matrix $D \in \mathbb{R}^{m \times m}$ such that all eigenvalues of $AD$ have negative real part ...
2
votes
1answer
35 views

Does changing rows in matrix changes column space order?

For example in matrix: $\begin{bmatrix} 0 & -2 & 3\\ 4 & 0 & 11\end{bmatrix}$ Column Spaces are asked. The answer is {(4,0), (0,-2)} Shouldn't we take the original matrix's ...
-2
votes
1answer
73 views

Using MATLAB to solve Poisson matrix equation

I'm not sure if this is the correct forum to post my question. I'm working on a Poisson-based maths assignment and am stuck as regards finding the solution to the Poisson matrix equation. The matrix ...
0
votes
1answer
36 views

Is it possible to define a Hessian Matrix for a Matrix-valued function?

So I'm doing a project on optimization (non-negative matrix factorization), which I know is not convex, from this question: Why does the non-negative matrix factorization problem non-convex? However ...
2
votes
2answers
55 views

Solving the matrix equation

How can I solve the matrix equation of the form $$ \mathbf{SXK} + \mathbf{X} = \mathbf{Y} $$ Here $\mathbf{S}$ and $\mathbf{K}$ are symmetric matrices, in addition $\mathbf{K}$ is a sparse symmetric ...
0
votes
1answer
19 views

Determine the principal strain of a 2x2 matrix

For a 2D problem the strain matrix is given by $$ \begin{bmatrix} \varepsilon_{xx} & \varepsilon_{xy} \\ \varepsilon_{xy} & \varepsilon_{yy} \\ \end{bmatrix} = \begin{bmatrix} 0 & 0.1 \\ ...
1
vote
1answer
44 views

Matrix differentiation by a vector in Least Squares method

In a book The Elements of Statistical Learning published by Springer we can find following statement: We can write $RSS(\beta) = (\mathbf{y}-\mathbf{X}\beta)^T(\mathbf{y}-\mathbf{X}\beta)$ ...
1
vote
0answers
35 views

Solving matrix equation containing trace and two-sided products

Let $A,B,Y$ be matrices and $d$ be a scalar. Is there an analytic solution for $A$ in the following expression? $$(A+B)^{-1} Y (A+B)^{-1} - (A+B)^{-1} + \frac{d A^{-2}}{\operatorname{tr}( A^{-1} + B^{...
2
votes
3answers
57 views

calculating eigen values from an equation

I'm trying to use this equation(in yellow) to calculate the eigen values of B = \begin{pmatrix} 1&1&1\\ 1&1&1\\ 1&1&1\\ \end{pmatrix} but I'm getting $$-λ^3+3λ^2-3λ$$ and the ...
0
votes
1answer
41 views

Completing matrix $B$ so ${B=PA}$

I have this problem. ${B=PA}$, where $P$ is a $l\times l$ invertible unknown matrix. $A$,$B$ are two $l \times m$ matrices. All entries of A are known. Some entries of B are known, but some ...
1
vote
1answer
40 views

Matrices Equations: is it Okay to Transpose Both Sides?

I have this matrices equation: $$(AX+I)^T=2I$$ Is it possible to transpose both sides to get this? $$ ((AX+I)^T)^T=2(I)^T \\ AX+I=2I$$ And then $$AX=I \\X=A^{-1} $$ Thank you.
1
vote
1answer
43 views

Is it true that $A^{T}P+PA>0$ for an unstable matrix $A$ and a positive definite $P$?

For a positive definite matrix $P$ and a matrix $A$ with all positive eigenvalues, how to guarantee that the matrix $Q=A^{T}P+PA$ is positive definite? I know if $A$ is a stable matrix (i.e. all the ...
0
votes
1answer
65 views

Which of the following statements is (are) true, for three matrices?

Let $A, B, C$ be three matrices such that $AB = C$. Which of the following statements is (are) true? the columns in $C^T$ are linear combinations of the columns in $B^T$ the columns in $C$ are ...
0
votes
1answer
58 views

Which of the following statements about linear system equations are correct?

Question: Which of the following statements about linear system equations are correct? Statements: A non-homogeneous system equations $Ax = b$ with $A$ of size $6\times7$ can have a ...
0
votes
1answer
33 views

Matrix solution to solve c1, c2, c3

Task: Determine which of the given numbers is the second coordinate of the vector x=[4; 5; 0] in the base {b1, b2, b3} where b1=...
0
votes
0answers
30 views

How to solve a system of matrix equations?

Given are two lists of 87 3x1 vectors each. I also know that when multiplying a 3x3 matrix by the nth vector of the first list, I get the nth vector of the second list. The 3x3 matrix is ​​the same in ...
0
votes
1answer
94 views

super commutation of matrices

Let $M_{p|q}(\mathbb{C}) = M_{p|q}(\mathbb{C})_0 \oplus M_{p|q}(\mathbb{C})_1$ be the super algebra of all $(p+q) \times (p+q)$ matrices. Let $A, B \in M_{p|q}(\mathbb{C})$ (not necessarily ...
0
votes
1answer
19 views

Values to whom the matrix is diagonalizable and invertible

I've got this matrix A=$$\begin{bmatrix} 2&0&3\\0&L&0\\4&0&0\end{bmatrix}$$ And I must to find the values for what this matrix is diagonalizable and they are 5 and 1 with ...