For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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0
votes
1answer
16 views

Efficient Row Sum of Factorized Matrix

I am currently computing the row sums of a reduced rank factored matrix by reconstructing a row subset of the original (approximated) matrix. The matrix was factored using SVD: A -> U, S, V -> U, SxV ...
-6
votes
1answer
47 views

The sum of invertible matrices is also invertible? [closed]

The sum of invertible matrices is also invertible ? THANKKSS!!!
0
votes
0answers
27 views

How to prove that a matrix lie group is (or is not) simple

I am having some difficulties in proving that some matrix lie groups are or are not simple. I am wondering what I should ask myself to solve this problem quickly. Should I look for normal subgroups? ...
1
vote
1answer
21 views

Changing the Form of this Factorisation

I'm brushing up on some high school maths and I'm currently revisiting determinants, specifically the factorisation of determinants. I'm working my way through a problem set and I keep getting stuck ...
1
vote
1answer
51 views

Matrix-vector product of a banded matrix

Suppose $A\in\mathbb{R}^{n\times n}$ is a banded matrix, i.e., a matrix with all of its nonzero elements on the main diagonal, i.e., $\alpha_{i,i}\neq 0$, the first superdiagonal, i.e., ...
0
votes
0answers
14 views

integral of trace function

How can I compute the below integral? $$ \int e^{-1/2*tr[(\mu\mu^T-\mu m^T-m\mu^T)\Sigma]}d\mu $$ in which $\mu \in R^{n},m \in R^{n},\Sigma \in R^{n*n}$ and $tr(.)$ is trace of matrix. I have ...
0
votes
0answers
36 views

The matrix with entries $M_{ij} = \frac1{t_i+t_j}$ is positive semidefinite

Prove matrix $M$, $n \times n$ is positive semidefinite if for $t_1 ,\dots,t_n > 0$: $M_{ij} = \frac1{t_i+t_j}$
2
votes
1answer
22 views

Does A+B symetric, A anti symetric, B symetric

I got $A^t = -A$ (A antisymetric) and B symetric: $B^t = B$. I need to know if $(A + B)^2$ is symetric. I couldn't find a formula which describe it. In addition, I know that A and B are non zero and ...
1
vote
1answer
27 views

How to Prove $V\otimes sl(k)=sl(V)$?

Let V be a vector space over a field $k$. Let $sl(n)$ be the set of all matrices elements from $k$ with trace zero. Is it true that $V\otimes _k sl(n)=sl(V)= \text{set of all $n\times$ n matrices ...
1
vote
1answer
48 views

Why the identity $P_X=P_XZ(Z'P_XZ)^{-1}Z'P_X$ with $P_X=X(X'X)^{-1}X'$?

Suppose $X$ and $Z$ are matrices such that $(X,Z)$ and $P_XZ$ both have full column ranks. Here, $P_X=X(X'X)^{-1}X'$. Consider a regression model $$ P_Xy=P_XZ\zeta+v\tag{A} $$ where OLS is used ...
-3
votes
0answers
20 views

non negative matrix factorisation [closed]

i am working on a project involving the use of non negative matrix (NMF) for the separation of a mixture of audio signals.can please give me the mathematical explanation of NMF so that i can ...
-5
votes
0answers
26 views

How to create a system of equations [closed]

How to create a system of equations delete
1
vote
0answers
17 views

What are the different ways of performing Triangular matrix-vector multiplication?

Suppose we have $$\left[\begin{array}{cccc} x_1 & 0 & 0 & 0 \\ x_2 & x_1 & 0 & 0 \\ x_3 & x_2 & x_1 & 0 \\ x_4 & x_3 & x_2 & x_1 \end{array}\right] ...
0
votes
1answer
33 views

Find unit vector given Roll, Pitch and Yaw

Is it possible to find the unit vector with: Roll € [-90 (banked to right), 90 (banked to left)], Pitch € [-90 (all the way down), 90 (all the way up)] Yaw € [0, 360 (N)] I calculated it without the ...
0
votes
0answers
39 views

A generalized eigenvalue problem

The generalized eigenvalue problem likes this: $\begin{pmatrix} 0 & C_{12}\\ C_{21} & 0 \end{pmatrix} \begin{pmatrix} \xi_1\\ \xi_2 \end{pmatrix}=\rho\begin{pmatrix} C_{11} & 0\\ 0 ...
0
votes
2answers
24 views

Cardinality of the set $S$ where $S=\{T:\Bbb{R}^3\to \Bbb{R}^3\mid T \text{ is a linear transformation with } T(1,0,1)=(1,2,3), T(1,2,3)=(1,0,1)\}$

Let $S=\{\,T\colon \mathbb{R}^3\to \mathbb{R}^3\mid T \text{ is a linear transformation with } T(1,0,1)=(1,2,3), T(1,2,3)=(1,0,1)\,\}$. Then $S$ is A. a singleton set B. a finite set containing ...
3
votes
1answer
25 views

Proving that $V = U \oplus W$ where $W$ and $U$ are sets of eigenvectors of $S: V \to V$

Let $V$ be a finite dimensional real vector space, $S : V \to V$ be a linear map such that $S^2 = I$. Show that $V = U \oplus W$ where $U = \{u \in V : Su = u\}$ and $W = \{ w \in V : Sw = -w\}$. ...
13
votes
0answers
234 views

Factorial of a matrix: what could be the use of it?

Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma ...
1
vote
0answers
30 views

Calculate Rotation and Translation Matrix to align elements of input matrix A to Target matrix B in 2d?

I have a matrix in 2D space; the matrix contains elements which I would like to translate into the center of the matrix. Then, I would like to rotate these elements (I mean the positions of the ...
1
vote
2answers
68 views

advanced solutions for an elementary problem!

Let $A\in M_3(\mathbb{R})$ be a matrix of rank one. Suppose that the first row of $A$ is an eigen vector of $A$. I want to show that $A$ is symmetric. My attemp: Actually its simple, for example ...
0
votes
1answer
27 views

Guass Jordan Elimination Matrix Problem

I followed the method as below: $ $[A]$ = \begin{bmatrix} 8 & 4 & 3 & | & 1 & 0 &0\\ 2 & 1 & 1 & | & 0 & 1 &0 \\ 1 ...
0
votes
1answer
22 views

Sum of component projection matrices

Show that if $X$ $=$ [$X_1$ $X_2]$ and $X_1'X_2 = 0$, then $P = P_1 + P_2$, where $P$ is defined as $X(X'X)^{-1}X'$, the projection matrix. Don't quite know where to start. I tried evaluating it by ...
0
votes
1answer
38 views

Prove that, considering $\mathbb{R}$ as an additive group, we have $SO_2(\mathbb{R})\cong \mathbb{R}/2\pi\mathbb{R}$ .

Let $SO_2(\mathbb{R})$ be the group of rotations of the circle under the operation of composition. Prove that, considering $\mathbb{R}$ as an additive group, we have $SO_2(\mathbb{R})\cong ...
0
votes
1answer
20 views

vector dot product

I have the multiple choice questions (from a past exam, not for marks don't worry) that states: If $u$ and $v$ are vectors such that $\| u+v \| = 2$ and $\| u-v \|= \sqrt{8}$, then the dot product of ...
0
votes
3answers
37 views

Is it possible to solve for values in a matrix such that all rows and columns have equal sum?

Is it possible to solve for values in a grid such that all rows have the same sum and all columns have the same sum where values in the table can be any real number? meaning: ...
1
vote
2answers
57 views

If I know that a matrix $G = (X^{T}X)^{-1}$, how can I recover what $X$ is?

If I have a matrix $G$ where I know that $G=(X^{T}X)^{-1}$, is there a way to find $X$? Specifically, I would like to find $G$ where $G$ is: $$G = \begin{bmatrix} 0.125 & 0 & 0 & 0 & ...
0
votes
1answer
18 views

exponential of elementary matrix $\exp(tE_{a,b})$

$E_{a,b}$ is the elementary $n\times n$ matrix with $1$ in $(a,b)$-entry and $0$ elsewhere. Compute $\exp(tE_{a,b})$ for $a$ not equal to $b$. If $a=b$ then they would be on the diagonal, so ...
0
votes
0answers
26 views

Possible largest number of column vectors with certain structure in a rank r matrix

My question is: If $A$ is a dimension $p$ symmetric square matrix with rank $r$ ($r<p$), and $a_{ij}$ is the element in the $i$-th row and $j$-th column. How many column vectors can satisfy ...
1
vote
2answers
83 views

How to take the inverse of the matrix $X^{T}X$, when it isn't invertible?

If I have a matrix $X$ and I am trying to compute $(X^{T}X)^{-1}$, which is the inverse of $X^{T}X$. However, each time I try to do it in some computing package like R, I get that $X^{T}X$ is ...
0
votes
2answers
36 views

Computing the inverse of $I - \lambda E$ where $E^{k+1} = 0 $ for some $k \geq 1$

If $E$ is a square matrix over $\mathbb{C}$ with $E^{k+1} = 0$ for some $k \geq 1$, then show that $I - \lambda E$ is invertible for all $\lambda \in \mathbb{C}$ by explicitly computing its ...
1
vote
1answer
40 views

$Z$ coordinates disappear in the general rotation transformation matrix.

I wanted to generate the general rotation transformation matrix ($3D$). But when I did the multiplication the result didn't include the original $Z$ coordinates,I don't know why the $Z$ disappeared. ...
0
votes
1answer
21 views

How to determine which of the 6 columns of this matrix are not linearly independent when combing with the rest?

I currently have a matrix $G$ with $6$ columns from a simulation that looks like: $$\begin{bmatrix}{} 1.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 & 0.0 ...
1
vote
1answer
22 views

Accelerating linear solve in MATLAB for a specific type of matrices

Inside a DG solver (so far 1D) I need to solve a linear system of equations multiple times. The order of the system is rather small ($N=10..20$). I need to solve the system $Ax=b$, where $A$ is the ...
1
vote
1answer
24 views

How to prove that determinant can take any real value using only this definition of the determinant?

I was reading some facts about the determinant and refreshed my memory with the fact that the determinant of the $ n\times n $ matrix can be defined as $ \det(A)=\sum_{\sigma \in S_n} sgn(\sigma) ...
0
votes
0answers
18 views

Finding Inverse of a matrix using elementary transformations

So I have to find the Inverse of A. $$ A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 4 \\ 3 & 4 & 3 \\ \end{bmatrix} $$ By using elementary row or column transformations.. The ...
2
votes
2answers
72 views

Why $x\in\ker A$ implies $x_i-x_j=\lambda \det A_{ij}$?

Suppose that $A$ is a real matrix with $n-2$ linearly independent rows and $n$ columns adding up to $0$. I can show that for any $x=(x_1,\dotsc,x_n)\in\ker A$ (that is, any $x\in\mathbb R^n$ ...
0
votes
0answers
11 views

Matrix notation: How would you apply a function to every column/row of a matrix?

Let's consider a real matrix A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ ...
0
votes
0answers
12 views

What is the effect on the spectrum by addition of a matrix with that of a rank 2 matrix?

Let $A$ and $B$ be two $n\times n$ matrices with rank of $B$ equal to $2$. Then how is the spectrum of $A$ and $A+B$ related? Or whether we can say something about - which of the eigenvalues of $A$ ...
0
votes
0answers
11 views

orthogonality condition for matrices and vectors

Given two orthogonal vectors in $n$ dimensions $\vec{x}$, $\vec{y}$ ($\vec{x} \cdot \vec{y} = 0$), what are linear transformations $T$ that will maintain orthogonality of $\vec{x}$ and $\vec{y}$? What ...
1
vote
0answers
36 views

Numerical Algorithm for $n \times n$ Matrix Inverse

I have to write a C program in which I have to compute the matrix inverse of a $n \times n$ matrix. Is there a convenient iterative process that I can use to do that? All I see is the co factor method ...
4
votes
1answer
49 views

Finding a matrix given eigenvalues and eigenvectors.

I am asked to construct a $4 \times 4$ symmetric matrix, with given eigenvalues and eigenvectors. I understand how to actually get $A$ as a product of $P^T, D$ and $P$, when $D$ is the diagonal ...
0
votes
0answers
22 views

What has been already done on spectrum of Hermitian matrices?

Could anyone suggest some books/articles related to the determination of eigenvalues and eigenvectors of some special complex Hermitian matrices?
0
votes
1answer
36 views

Under which condition does $Q>I_n$ result in $Q^2>I_n$?

Consider an $n\times n$ real matrix $Q>I_n$ (i.e., $Q-I_n$ is positive definite). Under which condition, $Q^2>I_n$ also holds? It is easy to show that if $Q$ is diagonalizable, $Q>I_n$ ...
0
votes
0answers
32 views

Solve system of two homogeneous first-order ordinary differential equa0ti0ns by eigenvectors. (7.16-1)

Please check my work and I shall have a few questions along the way. I am working out of the textbook Advanced Engineering Mathematics by Erwin Kreyszig, 1988, John Wiley & Sons. The problem to ...
0
votes
0answers
41 views

How to integrate over inverse of kernel Matrix [closed]

Let $x_i\in R^m, i \in \{1,2,...,n\}$ be a vector, $x_i(t)=\left( \begin{array}{c} x_{i1}\\ .\\ .\\ x_{ip-1}\\ t\\ x_{ip+1}\\ .\\ .\\ x_{im}\\ \end{array} \right)$, $x_i\neq x_j$ for $i\neq j$ and the ...
0
votes
1answer
14 views

Does pivot column include all entries within the column?

This is a quick fundamentals question. (maybe not even one) In linear algebra, a pivot column is a column where a pivot is located on. Does pivot column include all entries within the column even if ...
3
votes
0answers
55 views

What lies beyond the Möbius transform?

Consider the matrix $\pmatrix{a & b \\ c & d} ^n$ This is isomorphic to the $n$ th iteration of the Möbius transform $\frac{a z + b}{c z + d}$ when the determinant is nonzero. So I wonder ...
1
vote
0answers
20 views

Dimension and Basis of the $S_2$ set of symmetric matrices with $tr(A)=0, \forall A \in S_2$

For the following problem: Let $S_2$ be the set of symmetric matrices (with real entries) and zero trace. Prove that $S_2$ is a subspace of the space of all $M_{2\times2}$ matrices. ...
4
votes
3answers
258 views

Eigenvector and eigenvalue for exponential matrix

$X$ is a matrix. Let $v$ be an eigenvector of $e^{X}$ with corresponding eigenvalue $a$. Show that $v$ is also an eigenvector of $e^{X}$ with eigenvalue $e^{a}$ If $X$ is diagonalizable, then we can ...
0
votes
1answer
35 views

Exponential of matrix, taylor series

Compute $ exp(X) $for $X=$\begin{bmatrix}t&0\\0&s\end{bmatrix}, \begin{bmatrix}0&t\\-t&0\end{bmatrix} $ and $\begin{bmatrix}0&t\\t&0\end{bmatrix} The first part of the ...