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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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Minimising quadratic function subject to linear equality constraints.

The Problem At a higher level, I am trying to minimise a function that looks a bit like this \begin{equation} (x_2-x_1+c_1)^2 + (x_3-x_2+c_2)^2 + ... \end{equation} Subject to the constraint that $x_1+...
DBruwel's user avatar
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64 views

How to calculate the number of weeks gone, if I only have the end result of it? Simple question about Linear Algebra [closed]

The population from one week to the next is described by the following matrix: $$ \begin{pmatrix} 0 & 0 & 200 \\ 0,03 & 0 & 0 \\ 0 & 0,11 & 0,7 \end{pmatrix} $$ At the ...
Dan Louis's user avatar
-3 votes
0 answers
91 views

Given $2A-B$ and $A-2B$, find $\det(AB^{-1})$ [closed]

I've been stuck on this question for a very long time and I couldn't solve it. Please help. It's about matrices: $$2A-B=\begin{pmatrix}-5&3\\4&7\end{pmatrix},\quad A-2B=\begin{pmatrix}-5&4\...
Leonardo Schvartz Lenny's user avatar
-4 votes
0 answers
65 views

How is the multiplication of two matrices equal to the multiplication of their determinants? [closed]

Yesterday my lecturer gave the class an example to prove that the set M (that is the set of all $2\times 2$ matrices with $ad-bc ≠ 0$) under the multiplication operation is a group. My lecturer then ...
Ven's user avatar
  • 17
-1 votes
0 answers
65 views

Determinant of a matrix with 4 diagonals [closed]

How can I write the determinant below as nested sums? $\det \left[\begin{array}{cccccc} x_{1} & y_2 & z_3 & 0 & \cdots & 0\\ -1 & x_2 & y_3 & z_4 &...
GFP's user avatar
  • 9
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0 answers
68 views

Is there a closed form solution?

Question: Let $A\in\mathrm{GL}(n,\mathbb{R})[[u]]$ with $A=A^T$, that is a symmetric real matrix, which can not be decomposed into a blockmatrix with $0$ blocks, such that $$ A = \begin{pmatrix} A_1 &...
dForga's user avatar
  • 46
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0 answers
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representing matrix operations involving element wise multiplication

I have a vector $y$ of dimensions $t\times 1$ I have another matrix $x$ of dimension $t\times n$ I want to get a matrix $z$ of dimension $t\times n$ such that $z_{a,b} = x_{a,b} * y_a $ I then want to ...
dayum's user avatar
  • 204
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0 answers
29 views

Representation B,D [closed]

I am new here but thx for everyone for help. Here is some Linear Algebra problem for solving. Where we two polynomial bases B and D. And I read in Hefferon book some rule $$Rep_{B,D}(id) · Rep_B(~v) = ...
MBFG9000's user avatar
1 vote
1 answer
50 views

Iterative scheme to make matrice inverses of each other

Consider $X$ and $Y$, two invertible matrices which are nearly the inverse of each other. Feel free to ascribe any meaning to that... maybe $||X Y - I||_2 < \epsilon$ or maybe $||X^{-1} - Y||_2 &...
Arthur B.'s user avatar
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2 votes
2 answers
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If $[X,Y]=Y$ must $Y$ have $0$ as the only eigenvalue? [duplicate]

I was asking myself whether, if $Y\in M_n(\mathbb C)$ is such that $XY-YX=Y$ for some other $X\in M_n(\mathbb C)$, then the spectrum $\text{eig}(Y)$ must be $\{0\}$. I think that this is true for $n=1,...
wakewi's user avatar
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1 answer
25 views

For any given positive definite matrix, can we find a corresponding coordinate transformation?

Suppose that $\Omega\subset \mathbb{R}^3$ is a simply connected domain with $C^\infty$ boundary (for example a unit ball), and $A=A(x)$, $x\in\Omega$, is a $3\times 3$ positive definite matrix (may be ...
Jiawen Zhang's user avatar
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28 views

System of non-linear differential equations, $\dot{\vec{\theta}} = K^{-1} \hat{J}^{T} \vec{h}$

Suppose I have following system $\hat{J} \dot{\vec{\theta}} = \vec{h}$ with $\hat{J}$ is a function of $\theta$. I want to solve $\vec{\theta}$. Naively, one starts with the following construction. ...
phy_math's user avatar
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transformation of matrix equation $ \mathbf{y} =\mathbf{B} \mathbf{f}-\overline{\mathbf{B}} \mathbf{C} \mathbf{f}$

I found this transformation in a paper about image reconstruction: $$ \begin{aligned} \mathbf{y} & =\mathbf{B} \mathbf{f}-\overline{\mathbf{B}} \mathbf{C} \mathbf{f}=\left[\begin{array}{ll} \...
user1321902's user avatar
-1 votes
2 answers
89 views

How to solve $AX=B$ for $X$ if $A$ is not invertible? [closed]

How to determine the matrix $X$ that satisfies the following equation? $$ \begin{bmatrix} 5 & 5 & -3 \\ 2 & 2 & -1 \\ -1 & -1 & 1 \end{bmatrix} X=\begin{bmatrix} 10 & 2 &...
Mirna's user avatar
  • 19
1 vote
0 answers
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Bound an error of a matrix sequence given eigenvalues and eigenvectors basis.

Suppose $A\in\mathbb{R}^{n\times n}$ is a matrix with eigenvalues $\lambda_1,\ldots,\lambda_n$ such that $$|\lambda_1|>|\lambda_2|>|\lambda_3|\geq\cdots\geq|\lambda_n|$$ and its eigenvectors $v^{...
Fabrizio G's user avatar
  • 2,063
1 vote
1 answer
56 views

To prove or disprove $MM^t=\alpha I_{m\times m}$,where $M$ is an $m\times n$ matrix of rank $m$ with $m<n$.

Let $M$ be an $m\times n$ matrix of rank $m$ with $m<n$. If for some non-zero real number $\alpha$, we have $x^tMM^t x=\alpha x^tx$, for all $x\in \Bbb R^m$ then prove or disprove $MM^t=\alpha I_{m\...
Sandeep Tiwari's user avatar
0 votes
2 answers
44 views

Linear matrix equation with stacked matrices and repeated unknowns

I have the following seemingly simple linear matrix equation, where I want to solve for the unknown matrix $K \in \mathbb{R}^{p \times m}$: $$ \begin{bmatrix} K A_0 \\ K A_1 \\ \vdots \\ K A_n \end{...
user594147's user avatar
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0 answers
38 views

Given a singular matrix $B$ and a result $C=A\times B$, find matrix $A$ over finite fields.

The problem is a conituation of this problem, but over finite fields. In the context of a finite integer field, particularly when all entries in matrices $A, B$, and $C$ are drawn from a finite ...
X.H. Yue's user avatar
3 votes
1 answer
79 views

Find the determinant of the linear transformation $T(f) = 7f-3f^{'}+7f^{''}$ from the space $V$ spanned by $\cos x$ and $\sin x$ to $V$

Find the determinant of the linear transformation $T(f) = 7f-3f^{'}+7f^{''}$ from the space $V$ spanned by $\cos (x)$ and $\sin (x)$ to $V$. Ok, so I've gone through the motions of replacing $f$ for $...
hiitslight's user avatar
5 votes
3 answers
109 views

$ABACA = 0 \Longrightarrow BAC = 0$ if $A,B,C \ge 0$ are symmetric.

Problem. $A, B, C$ are $n \times n$ symmetric positively semi-definite matrices. Prove that $ABACA = 0 \Longrightarrow BAC = 0$ if $A,B,C \ge 0$ are symmetric. My attemp (there's mistake in it). We ...
Sergei Nikolaev's user avatar
4 votes
1 answer
61 views

Binary matrix power for a specific entry.

Consider a square $n\times n$ matrix $A$ whose entries are binary, that is, for all $i, j\in [n]$, it holds that $A_{i, j} \in \{ 0, 1\}$. I am interested in the following decision procedure: Given a ...
Bader Abu Radi's user avatar
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0 answers
37 views

Is there any mathematical results stating when there are 0's in the inverse of a square matrix given 0's in the original matrix?

I am working with square invertible matrices. Denote the n-by-n matrix as $A \in \mathbb{R}^{n \times n}$. Say we know that there are some 0's in the matrix. For instance: $A_{ij} = 0$ for some $i,j$ ...
ajl123's user avatar
  • 183
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55 views

Ways to invert complicated matrix formulas

I have two somewhat complicated matrix formulas that convert the mean vector and covariance matrix for a certain variable, $\mu \in \mathbb{R}^n$ and $\Sigma \in \mathbb{R}^{n \times n}$, into the ...
dherrera's user avatar
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2 answers
53 views

Geometry Application for System of Linear Equations

Question : Find the intersection (if any) of the line $x=(1,0,-1)+\lambda(3,2,1)$ and the plane $x = (-1,-7,-7) + \alpha(3,5,1)+ \beta(1,-2,-5)$. My work so far: I equated both equations and ...
N.K's user avatar
  • 3
1 vote
1 answer
61 views

Reformulate a system of ODEs into an ODE on some norm

I had a complicate system of ODEs which I don't expect to obtain an explicit solution, while the ODE on their sum seems to be promising to have an explicit solution. $$\frac{d\sum_{i=1}^n x_i(t)}{\...
Silentmovie's user avatar
1 vote
1 answer
39 views

How can I find a solution to this matrix equation?

$$D=X-M(X\circ P)$$ Solve for $X$ where $D, X, P$ are vectors of size $n$. $M$ is a matrix of size $n \times n$. $X\circ P$ indicates element wise multiplication of $X$ and $P$ such that $(X \circ P)...
Chechy Levas's user avatar
1 vote
0 answers
89 views

If $A+B=AB$, Does it imply $A$, $B$ Commute?

If $A+B=AB$, Does it imply $A$, $B$ Commute? What I did is $$A+B=AB \Rightarrow I-A-B+AB=I \Rightarrow (I-A)(I-B)=I$$ Since $I-A, I-B$ are inverses of each other we have $$(I-B)(I-A)=I \Rightarrow B+...
LifeIsMath's user avatar
4 votes
1 answer
94 views

Solving a certain matrix equation

I am trying to solve the (real) matrix equation $X^{-1}AX=Y$, where the matrix $A_{n\times n}$ is given and the matrices $X_{n\times n}, Y_{n\times n}$ are unknown matrices, with the only restriction ...
Ali N's user avatar
  • 91
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0 answers
51 views

A special property of a system of linear equations in n > 3 dimensions

The origin of the question is physics, so I have to explain the idea behind it. In three dimensions the Lorentz force $F$ of a magnetic field $B$ on a particle (with charge $q = 1$) with velocity $v$ ...
TomS's user avatar
  • 259
0 votes
1 answer
55 views

Matrix multiplications with SVD

I'm trying to understand the calculation of $SU = U(\sigma^2 I + D^2)$, which I need to prove with the condition $S(\sigma^2 I + WW^\top)^{-1}W = W$. Let $W \in \mathbb{R}^{d \times m}$ and $W = UDV^\...
Lopsio's user avatar
  • 85
0 votes
1 answer
32 views

Correspondence of forms of the same solution of a system of linear equations

I am currently reading Mathematics for Machine Learning book (p.27-28, freely available). And I am really confused with how two different forms of a solution of a system of linear equations correspond ...
Egor's user avatar
  • 3
0 votes
2 answers
31 views

Specific value for an invertible matrix

I have been given the following matrix, and have been tasked with finding the values of "a" that makes it invertible. I know that for a matrix to be invertible, then the determinant of said ...
Markus J's user avatar
1 vote
1 answer
27 views

Integer Linear Programming - Dividing n people into m groups of specific sizes

I've recently asked this question about dividing n people into m groups for the specific model I used to solve the assignment problem of dividing the people into groups (boolean variables xij that ...
Zufra's user avatar
  • 197
0 votes
0 answers
33 views

Invertibility of matrix exponential

I have found several proofs that matrix exponentials are invertible. However, in Matlab, I have a matrix $A \in \mathbb{C}^{100}$ that is full rank. However, ...
user3284182's user avatar
2 votes
1 answer
20 views

Equation for Counting Unique RREF "Cases" for any (m x n) Matrix

Given a real matrix $A$ of size $(m \times n)$, I am seeking to determine the number of unique reduced row echelon form (RREF) "Cases" that exist for a given matrix size. Two RREF matrices ...
FaffyWaffles's user avatar
0 votes
1 answer
50 views

How to find the rank of matrix $A$?

This question is from my assignment in linear algebra and I am not able to make any significant progress on this. Question: Let $A$ be a $5\times 4$ matrix with real entries such that $Ax=0$ iff $x=0$...
user avatar
1 vote
1 answer
33 views

what is wrong with my equivalent transformation on optimization below

$\mathbf{R_1}$ and $\mathbf{R_2}$ are positive definite symmetric matrices, $\mathbf{q}\in\mathbb{C}^{N\times1}$, and I want to maximize the objective function below $$ J(\mathbf{q})=\frac{\mathbf{q}^{...
lei zhou's user avatar
2 votes
1 answer
108 views

What is the correct method to find the intersection of these subspaces?

I have to find the intersection in $\mathbb R^4$ of the subspaces generated by these generators: $$S1 = span[(1 ,2, 2, 1)^T,(0, 1 ,0, -1)^T]$$ $$S2 = span[(1, 0 ,1, 2)^T,(1 ,2, 1 ,0)^T]$$ I wrote ...
ssj's user avatar
  • 21
2 votes
1 answer
59 views

Integer Linear Programming - Dividing n people into m groups

I have modeled the problem of dividing n people into m groups using a binary $nxn$ matrix that we will call X. If $x_{ij} = 1$ it means that person i is with person j in the solution's groups. If $x_{...
Zufra's user avatar
  • 197
1 vote
1 answer
42 views

Proving Row Independence in Matrix Product Resulting from Linearly Independent Vectors and Permutations

Let's denote by $u_1, u_2, \ldots, u_n$ a set of linearly independent vectors. For each $i$ from $1$ to $n$, we define $\sigma_i$ as distinct permutations of the numbers $1$ to $n$. Construct matrix $...
Fernand's user avatar
  • 15
0 votes
0 answers
15 views

If a singular square matrix is multiplied by a non-singular square matrix, the null space of the result is what?

If a non-singular square matrix $B$ is multiplied by a singular square matrix $A$ of the same order, the nullspace of the resulting matrix $C=B\times A$ or $C'=A \times B$ remains unchanged from that ...
X.H. Yue's user avatar
2 votes
1 answer
38 views

if you have $mn + m+n$ variables and m+n equations involving, how do you know that you can omit an equation to solve for $m+n-1$ variables?

I'm reading this paper and I'm wondering how what I asked above is intuitively true, based on the equations shown below. Let the total product shipped from the $i^{th}$ factory to the $n$ cities be $...
John van Zalk's user avatar
-2 votes
1 answer
29 views

If A is idempotent then A+AB−ABA isidempotent for any square matrix B with the same dimension as A. [closed]

If A is idempotent then A+AB−ABA is idempotent for any square matrix B with the same dimension as A. I have this question to solve and I tried squaring the entire expression and then simplifying it ...
Valentina Tanguy's user avatar
0 votes
1 answer
35 views

Any vector can be obtained by rotation and scaling from a unit vector?

How can i justify rigorously that for some vector $x$ there is a unitary matrix $R$ such that $x=\|x\|Ry$ where $y$ is some normed vector. This intuitively very clear to me but somewhat I have no idea ...
Perelman's user avatar
  • 259
0 votes
0 answers
25 views

Determinant of an $n\times n$ matrix where $n$ is unknown [duplicate]

I´m trying to solve the following problem: Let $a$ be a scalar. Consider the set $\left\{ v_{1},v_{2},...,v_{n} \right\} \subseteq \mathbb{R}^{n}$ such that the $i$th element of $v_{i}$ is equal to 1 ...
gnzlama's user avatar
  • 75
0 votes
0 answers
32 views

Is the value of the determinant of a large matrix the same as the value of the determinant of its chunked matrix?

An $8 \times 8$ matrix $M$ is directly computed by computing its determinant $|M|$, and by dividing it into blocks $ M = \left [ \begin{matrix} A & B \\ C & D \end{matrix} \right ] _{4 \...
Lucky's user avatar
  • 33
2 votes
3 answers
125 views

Antiderivative of a linear matrix expression

Let $f:\mathbb{R}^{n\times m}\rightarrow \mathbb {R}$ be a function that takes an $n\times m$ matrix $X$ and maps it to the real line. Suppose that the derivative of $f$ with respect to one element $...
user_lambda's user avatar
  • 1,400
0 votes
1 answer
43 views

Inverse $T$ matrix of a 3*3 matrix. [closed]

I have the matrix $$ A= \begin{bmatrix} 1 & 4 & 1\\ 0 & 2 & 5\\ 0 & 0 & 5 \end{bmatrix}. $$ I have found the $$ T= \begin{bmatrix} 1 & 4 & 1\\ 0 & 0 & 1\\ 0 &...
Irini's user avatar
  • 9
2 votes
1 answer
34 views

Find a rotation matrix given some constaints on points transformation

I'm looking for elegant way to find a rotation matrix between RefA and RefB with all the points known in RefA and a 6 constraints (corresponding to a solid transformation with 6doF) in RefB. I set as ...
Ronan's user avatar
  • 21
0 votes
1 answer
13 views

The definition of the conditional number for a matrix

According to https://en.wikipedia.org/wiki/Condition_number, the conditional number for the matrix $M$ is $||A|| \times ||A^{-1}|| = ||A A^{-1}|| = 1$. In another word, is $||A||$ the absolute value ...
CPW's user avatar
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