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

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0
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0answers
9 views

Matrix function to express pair-wise distances of rows in $X, Y$

There are two real matrices: $X, Y$ with $X$ being of dimension $n_1$ x $p$, $Y$ of dimension $n_2$ x $p$. The goal is to form the matrix $D$ of dimension $n_1$ x $n_2$ where each element $d_{ij}$ ...
1
vote
1answer
28 views

Derivations on matrix algebra

Let $M=M_2(\mathbb{C})$ and let $\delta:M \mapsto M$ be a $\mathbb{C}-$linear map such that $\forall a,b \in M$ we have $\delta(ab)=\delta(a)b+a\delta(b)$. Prove that $\delta$ is of the form ...
-1
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0answers
22 views

find eigenvalues and eigenvectors of A3X3 [on hold]

Find the eigenvalue and eigenvectors for the matrix A 3x3 A= 1 a a a 1 b a b 1
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0answers
9 views

Simple Matrix Equivalence

Let R be an N X T matrix, 1 a T X 1 matrix of ones, m an N X 1 matrix such that m = $\frac{1}{T} * R * 1$, I is a T X T identity matrix. Now, define V = $(\frac{1}{T}) * (R - m1') * (R - m1')'$ I ...
2
votes
1answer
29 views

Solve for matrix that is hidden inside a scalar

Let $X$ an invertible $n\times n$ matrix, parameter vectors $P, K$ $n\times 1$, vector $\Omega$, $1\times k$ and matrix $\Lambda$ $n\times k$. I would like to solve with respect to $X$ the equation: ...
3
votes
3answers
49 views

How can we calculate the exponential form of a rotation matrix

Considering the rotation matrix: $$ A(\theta) = \left( \begin{array}{cc} \cos\space\theta & -\sin \space\theta \\ \sin \space\theta & \cos\space\theta \\ \end{array} \right) $$ How can I ...
1
vote
2answers
29 views

How many solutions exist for a matrix equation $A^2=I$?

Let $A$ be a square matrix of order three or two, and $I$ be a unit matrix. How many solutions are possible for the equation $$A^2=I$$? In case the solutions are infinite, or very large, how do I ...
0
votes
1answer
14 views

Identity matrix addition and inverse matrices

I am trying to reduce the following: x and y column vectors yt is the transposed column vector $(I - \frac{1}{(1+ y^t x)} * x y^t) (I + x y^t) = I$ I am stuck at $x y^t * y^t X = x y^t (x y^t +I)$ ...
2
votes
2answers
42 views

Properties of the matrix square root

In a paper I am reading, it is claimed that if $A, B \in \mathbb{R}^{n \times n}$ are positive definite, then $$ A^{1/2} (A^{−1/2} B A^{−1/2})^{1/2} A^{1/2} = A (A^{-1}B)^{1/2} $$ because of the ...
0
votes
1answer
25 views

Next step to show that these matrice expressions are equal?

This is a problem from Discrete Mathematics and its Applications I know invertible means it is possible to take the inverse of this matrix. This is definition of a power of a square matrix from my ...
0
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1answer
20 views

Find all solutions to $Bx =[7, -10, 7, 0]^T$

$$ B=\left[ \begin{array} k1 & 0 & 2 & 1\\ -3 & 2 &-1 & 5\\ 2 & -1 & 1 & 4\\ 0 & 3 & 2 & 4\\ \end{array} \right] ...
0
votes
2answers
63 views

Solving augmented linear system $\left(\begin{smallmatrix}1&0&-2&~~~&0\\0&1&0&&0\\0&0&0&&0\end{smallmatrix}\right)$

$$\left(\begin{smallmatrix}1&0&-2&~~~&0\\0&1&0&&0\\0&0&0&&0\end{smallmatrix}\right)\to ...
3
votes
3answers
42 views

Derivative of $tr((AX)^tAX)$

I'm trying to calculate the derivative (with respect to the matrix $X$) of the function $f(X) = tr((AX)^t(AX))$, Chain's rule gives that $\nabla_X(f(X))=\nabla_X(tr(AX))\nabla_x(AX)$ However I'm ...
2
votes
0answers
44 views

Determinant of a sum

We have that: $\textbf Y \in \mathbb{R}^{n \times q}, \textbf G \in \mathbb{R}^{n \times n}, \textbf P \in \mathbb{R}^{n \times n}, \textbf Q \in \mathbb{R}^{q \times q}$. Furthermore, $\textbf G$ is ...
2
votes
1answer
35 views

How to solve $C = X^\top C X$?

All matrices are $n \times n$. $C$ is real symmetric positive definite. How to solve $C = X^\top C X$ for $X$? I am interested in characterizing both the set of real matrices satisfying the equation ...
0
votes
2answers
37 views

What is the relationship between matrix position to array index of corresponding matrix?

I want to know the exact relationship between matrix position and array index, where array contains the matrix data in each row appended format. For example: I had a matrix of $3 \times4$ as follows: ...
0
votes
2answers
8 views

Matrix location by indices?

The formula for location number by successive rows for an $n$ by $m$ matrix is $f(x,y)=m(x-1)+y$ That is, for example a matrix with $n=3, m=4$ \begin{bmatrix} 1 & 2 & 3 & 4\\ ...
-2
votes
1answer
41 views

Linear algebra proving question, matrix algebra [closed]

I am stuck on this question and I don't even know how to start.Attached below is the picture of the question
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1answer
60 views

Looking for fast computation method of $Ax=b$ ($A$ is sparse matrix)

I am looking for fast method to solve linear equation $$Ax=b$$ In which A is sparse matrix. Could you suggest to me some current method for this task. Thank in advance
0
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1answer
35 views

transform xWz to Wxz using tensor product

The equation I need to solve is $\mathbf{R} = \mathbf{X}^T\mathbf{W}\mathbf{Y}$ where $\mathbf{R}$ is in $\mathbb{R}^{l \times m}$. $\mathbf{X}$, $\mathbf{W}$, and $\mathbf{Y}$ are in ...
0
votes
1answer
30 views

Converting Quaternion or 4x4 Matrix to 3x3 Matrix representation.

I'm working on some code that manipulates an Axis-Aligned Bounding Box, so it always encompasses the object it borders. I use a 3x3 matrix to re-size the box when it rotates. The only issue is I only ...
0
votes
0answers
38 views

Equation with unknown with high pow value.

Please someone help me to isolate $b$ from the rest of equation where $d$ and $n$ are known values. $d$ and $n$ can be different in different cases. $$ nb - b^{d} = n + 1 $$
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3answers
33 views

How to solve for X matrix X*A + X = 2B

I am given homework where I have to solve for matrix $X$ equation $$B^{-1}XA = -B^{-1}X + 2E,$$ where $A$ and $B$ are known matrices and $E$ is the identity matrix. I simplified it in the following ...
0
votes
1answer
29 views

How to change this formula into matrix form

In MATLAB, matrix computation is very efficient. Thus, we should change the computation into matrix forms as far as possible. I have the following formula: $$ \frac{\partial L}{\partial z_j} = ...
0
votes
1answer
19 views

Matrix representation from a linear function

I'm studying an opinion formation model [1] where the main rule is: $p^{(t)}_i = \frac{p^{(0)}_i + \sum_{j \in N(i)} w_{i,j} p^{(t-1)}_j}{1 + \sum_{j \in N(i)} w_{i,j}}$ Now, I'd like to represent ...
0
votes
2answers
29 views

Find matrix such that:

Find matrix such that: $$\begin{pmatrix} 3 & 2 & 3 \\ 3 & 6 & 3 \\ 1 & 2 & 4 \end{pmatrix} X+ \begin{pmatrix} 1 & 1 & 0 \\ 1 & -1 & 2 \\ 1 & 0 ...
1
vote
2answers
69 views

Simulating simultaneous rotation of an object about a fixed origin given limited resources.

Sorry if the title is a bit cryptic. It's the best I could come up with. First of all, this question is related to another question I posted here, but that question wasn't posed correctly and ended ...
1
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0answers
49 views

How do I solve this matrix?

How do I expand the following matrix? I have no experience in linear algebra (well 35 years ago..) Its the equation just after equation 16b in this link Here are the equations in question: $$ ...
2
votes
1answer
104 views

identity of $(I-z^nT^n)^{-1} =\frac{1}{n}[(I-zT)^{-1}+(I-wzT)^{-1}+…+(I-w^{n-1}zT)^{-1}]$

I am trying to understand the identity $$(I-z^nT^n)^{-1} =\frac{1}{n}[(I-zT)^{-1}+(I-wzT)^{-1}+...+(I-w^{n-1}zT)^{-1}] \quad (*),$$ where $T \in \mathbb{C}^{n\times n},z\in \mathbb{C}$ and the ...
0
votes
2answers
24 views

How to derive the following matrix equation?

The SVD of a symmetric matrix $X$ is as follows: $$ [U, \Lambda, U^T] = svd(X); $$ then, if we have the following matrix: $$ Y = {\left(U(\Lambda+\alpha I)U^T\right)}^{-1} U \Lambda $$ My question is: ...
1
vote
1answer
18 views

Qestion about Eigenvector, basis for the solution

I'm confused with some question currently I'm trying to solve. If you help that will be grateful. Given the matrix find eigenvalues and eigenvectors $$ A = \begin{bmatrix} 4 & -2 ...
0
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0answers
36 views

what $A$ makes $X$ in $\dot{X}=-AX$ converge?

Suppose that $X,A$ are all $n\times n$ matrix, $X$ is variable and $A$ is constant. We have a differential equation $\dot{X}=-AX$, then my question: what conditions should $A$ satisfies to make $X$ ...
0
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0answers
16 views

How do I generate a Spotlight Projection Matrix for Shadow Mapping?

I'm currently in the Process of making a simple little shaodw mapping ystem for Ogre3d. I'm currently stuck at Mapping the Shadow map texture to the object for depth comparison because I have no idea ...
0
votes
0answers
21 views

Proof of inverse matrix element explicit formula [duplicate]

There is a matrix: A, and exists an inverse matrix: A^-1 which elements are b. (b)ij = adj(Aji) / det(A) What is the proof of this equation?
3
votes
2answers
85 views

Matrix equation $aX^{3} + bX^{2} = I$.

I want to solve the matrix equation for $X$ $$aX^{3} + bX^{2} = I,$$ where $a,b \in \mathbb{R}$ and $X \in \mathbb{R}^{n\times n}$. My thoughts: If $a = 0$ or $b = 0$, the solution is easy. If $a, ...
0
votes
0answers
27 views

Properties of log map on matrices in $SE(3)$

I am learning about the log map on $SE(3)$ and I want to check my understanding of properties for use in solving an equation. Are the following true, for A, B, C as elements of $SE(3)$? $$ \log(ABC) ...
1
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2answers
60 views

How to adopt the Woodbury matrix identity to this matrix formula

The Woodbury matrix identity is defined as follows: $$ {(A+UCV)}^{-1}=A^{-1}-A^{-1}U{(C^{-1}+VA^{-1}U)}^{-1}VA^{-1} $$ I want to use the Woodbury matrix identity theorem to change the following ...
1
vote
1answer
43 views

Find unknown matrix in matrix equation

Given a matrix $A$ and a symmetric positive definite matrix $Y$, find a symmetric positive definite matrix $X$ which solves $$ X + AXA'+A^2X\left(A'\right)^2=Y $$ (This differs from the algebraic ...
2
votes
1answer
42 views

Stochastic matrix A^n=const.

Let $A\in M(n,n)$ be a doubly stochastic matrix such that $$ A^k=\begin{pmatrix} 1/n & \cdots & 1/n\\ \vdots & \ddots & \vdots\\ 1/n & \cdots & 1/n\\ \end{pmatrix} $$ for some ...
1
vote
2answers
87 views

Simultaneously diagonalization of two matrices.

Let $A$ be a real symmetric matrix and $B$ a real positive-definite matrix. Is it possible to simultaneously diagonalize of $A$ and $B$? Thank you very much.
4
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1answer
69 views

Matrix equation solution

Does anybody know how to solve this matrix equation: $$P = P P^T R + X,$$ where $P, R,$ and $X$ are vectors with $n$ elements, and $P$ is the unknown vector?
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0answers
26 views

Ring structure of tuples mod k

Consider a vector of n integers $$A= a_1, a_2, ... a_n$$ Such that for another vector $$B= b_1,b_2... b_n$$ $$AB^T \equiv 0 \mod k$$ For an integer k. I was playing around with these structures ...
0
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1answer
51 views

Solving eqn. of the form K = AGL + BGT, where A,B,L,T are invertible matrices.

I am obtaining the following equation in a regression problem: \begin{eqnarray} Z'_1Y_1\Omega^{-1}_{1}A+Z'_2Y_2\Omega^{-1}_{2}A = Z_{1}'Z_1\Pi A'\Omega^{-1}_1A + Z_{2}'Z_2\Pi A'\Omega^{-1}_2A ...
0
votes
1answer
29 views

Linear transformation to higher dimensional space.

There is a 7-by-6 matrix $H$ given. Its rank is 6. I'd like to design a 6-by-5 matrix $D$ such that the following holds: $ \left[ \begin{array}{l} l_1(a_1, a_2, a_3, a_4) \\ l_2(a_1, a_2, a_3, a_4) ...
0
votes
2answers
44 views

how to differential exponential of a matrix variable $f(X)=e^{X(t)\mathrm{d}t}$?

I have a function about a square matrix $X$ which depends also time: $f(X)=e^{X(t)}$, $t$ is time. So how to differential it about time to have $\frac{\mathrm{d}f(X)}{\mathrm{d}t}$? I know that ...
0
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0answers
31 views

What is the proof of this? (Matrices, Pivot)

I have a matrix : $A$ I pivoted $A$ with a pivot element $(p)$ and I get this matrix: $B$ What is the proof of this equation? $|A|$ = $\frac{1}{p}. |B|$
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1answer
24 views

Transforming a square matrix A into B

Let's say I have $A= \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{bmatrix}$ and $B= \begin{bmatrix} b_{11} & ...
0
votes
2answers
20 views

Solving systems of equations using matrices by row reduction

Solve the following system for $a$, $b$, and $c$: $$\begin{pmatrix}1 & -1 & 2\\2 & -2 & 2 \\ 3 & -3 & 2\end{pmatrix}\begin{pmatrix}a\\b\\c\end{pmatrix} = ...
3
votes
2answers
69 views

The basis of a matrix representation

If I have the linear map $f:\Bbb{R}^n\rightarrow \Bbb{R}^m$ then we can write $f$ as like the following: $$f\left(\vec x\right)=A\vec x$$ Where $A$ is a matrix. I think $A$ is called the standard ...
0
votes
2answers
63 views

Solution to a system of linear equations with an unknown matrix product

Consider the system of equations $$ Xy=Ab $$ where $X$ and $A$ are $m \times m$ invertible matrices and $y$ and $b$ are $m \times n$ matrices. The matrices $X$ and $y$ are unknown and the matrices $A$ ...