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
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0answers
35 views

Determinant of block matrices of block matrices with different dimensions

Please, how can I find the determinant of the following matrix? $$P=\left( \begin{array} {c,c} A \quad B \\ C \quad 0 \end{array} \right)$$ where $A$ is a 2x2 block matrix, $B$ is a 2x1 matrix, $C$ ...
8
votes
1answer
66 views

Prove a generator of $\mathrm{SL}_2(\mathbb{R})$

Definitions Let $\mathrm{SL}_2(\mathbb{R}) := \left \{M := \begin{pmatrix} a & b\\ c & d\end{pmatrix} \in \mathbb{R}^{2 \times 2}: \det{M} = 1 \right \}$ with matrix multiplication be a group ...
1
vote
1answer
68 views

Perron-Frobenius Theorem. A particular case?

Let $\{a_{i,j}\} =A \in \mathbb{R}^{N \times N}$ be a non-negative matrix, such that: $a_{i,i} = 0 ~~ \forall i \in \{1, \ldots, N\}$ $a_{i,j} \geq 0 ~~ \forall i \neq j$ Given the previous ...
5
votes
1answer
134 views

Extended matrix function

I have a continuous matrix-valued function $f:\mathbb{R}^d\mapsto {\cal M}_{k\times d}$, with $d<k$, such that $f(x)$ is full rank for all $x\in\mathbb{R}^k$. Can I extend this function to be a ...
0
votes
0answers
12 views

composition of multiple scales

i want to scale a point multiple times. Let P(x, y) be the point that is to be scaled, S the scale matrix and T the translation matrix. To scale the point P according to a point Q(x,y) you have to do ...
1
vote
4answers
264 views

Algebra-sum of entries in each column of a sqaure matrix = constant

This is a question from an algebra homework and I am just looking for some tips. The question is: We have: $M$: an $n\times n$ matrix with real entries $c$: a real constant the ...
1
vote
1answer
50 views

Exploiting structure of linear equations to solve them

So I have a bunch of linear equations $Ax=y : A \in R^{m,m}, y \in R^{m}$. Note that $A$ is a square matrix. The question is if I can decompose $A$ as $$A = D + uv^T$$ where $D$ is diagonal, ...
0
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0answers
45 views

Determinant of a sum of matrices

Does anyone have a square matrix $A$ such that $A$ is invertible, $\det(A+B)$ is easy to calculate for all invertible matrices $B$. $${}$$
6
votes
0answers
164 views

Matrix diagonalization theorems and counterexamples: reference-request.

I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra. In this question I understand the question of matrix diagonalization very broadly: suppose we have ...
1
vote
1answer
154 views

Principal EigenVector of an Adjacency matrix of an undirected graph

For an undirected graph, since the adjacency matrix will be symmetric, can we draw any relations between the principal eigenvector and the degree of nodes in the graph. Also can we do the same with ...
1
vote
1answer
71 views

Motivation for Conjugate transpose of a matrix

I'am currently going through a self study of Linear algebra . I'am finding it difficult to grasp the intuition behind the concept of Conjugate transpose of a matrix .Why take the complex conjugate of ...
2
votes
1answer
39 views

Is $GL(n;\mathbb{C})$ algebraic or not?

The set of $n\times n$ matrices can be identified with $\mathbb{C}^{n^2}$. 1) Consider the subset $V$ of the affine space $\mathbb{A}^{n^2+1}$ (note plus one) given by $$V:=\{(x_{ij},t): ...
0
votes
1answer
40 views

What $T(2x^3 -x^2 - x)$ is?

Let $B = \{p_1=x^3+1, p_2=x^3-1, p_3=x^2 + 2x, p_4=2x^2 +3\}$ a basis for $\mathbb{R}^3$. Let $T:\mathbb{R}{\left[ x \right]_3}\rightarrow \mathbb{R}{\left[ x \right]_3}$, such that: $T(p_1) = 3x^2 ...
7
votes
1answer
132 views

How to prove this $A$ is an invertible matrix

let Symmetric matrix $A=(a_{ij})_{n\times n},n\ge 2$,and $$\begin{cases} a_{jk}=j+k\cdot i&j< k\\ a_{jj}=2j\cdot(i+1) \end{cases}$$ where $i^2=-1$ show that :$A$ is Invertible matrix My ...
2
votes
1answer
42 views

Relationship of eigenvalue/eigenvector of hermitian matrix R and QRQ (Q is diagonal)

For a hermitian matrix R and a diagonal one Q, is there any relationship between eigenvalues/eigenvectors of R and QRQ? To be specific, assuming the eigenvalue decomposition of R is R=VDV*, then can ...
1
vote
1answer
38 views

Help with load balancing math based on fractional capacity

I'm looking to create an algorithm that allows me to select a number(index) from a list based on it's fractional weight component. It's for load balancing, I'll give an example below of what I mean. ...
1
vote
1answer
33 views

Meaning of off-diagonal multivariate covariance matrices

My terminology might be a bit sloppy. I apologize in advance. I'm reading on multivariate probabilistic distributions, particularly on Gaussian normal distribution (in the context of probabilistic ...
2
votes
2answers
53 views

Hessian of Morse function on $S^{n}$ mistake

I am trying to get that $f(x_{0},...,x_{n+1})=x_{n+1}$ has $index_{(0,...,0,1)}=n$ Can you find my mistake or post a partial solution? My attempt I evaluate df using inverse of stereographic ...
2
votes
1answer
94 views

Texture mapping from a camera image (knowing the camera pose)

I'm not sure if I should ask this question here or on stackoverflow, so forgive me if I'm wrong. I want to apply a texture (taken from a camera) on a 3D surface, let me explain my problem: I have ...
1
vote
1answer
114 views

Eigenvalues of a $3\times3$ orthogonal matrix

Can anyone give me an example of 3x3 orthogonal matrix with complex eigenvalue.
0
votes
1answer
42 views

What do real eigenvalues imply for a matrix

Suppose we have a matrix $A \in \mathbb{R}^{n \times n}$ with $\textrm{eig}(A)=\{ \lambda_1, \lambda_2, \ldots, \lambda_n\}$ such that $\lambda_i \in \mathbb{R}$. Does the realness of the eigenvalues ...
2
votes
1answer
66 views

A Matrix Optimization Problem

Given an $n\times d$ matrix $Y$, I am looking for an algorithm to find an $n$-vector $\mathbf{v}$ ($0\le \mathbf{v}_i\le 1$ for all $i$) that minimizes $\sum_{i:X_i<0}X_i$, where $X:= \mathbf{v} ...
1
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2answers
42 views

If we add $I$ to a matrix $M$, does that mean we always add 1 to each of $M$'s eigenvalues?

Title says it all, Suppose we have a matrix $\mathbf{M} \in \mathbb{R}^{N \ \text{x} \ N}$, with eigenvalues $\lambda_i$, for $\ i = 1, 2 ... N$. If we now add the identity matrix $\mathbf{I}$ to ...
0
votes
1answer
31 views

Product rule type formula for $\nabla \cdot (M(x)v(x))$ where $M(x)$ is a matrix and $v(x)$ is a vector?

Let $M(x)$ be a $n\times n$ matrix with each element depending on $x$ a variable on $\mathbb{R}^n$. Let $v(x)$ be a vector. Is there a nice product rule formula for $\nabla \cdot (M(x)v(x))$?
0
votes
1answer
17 views

Linear transformation over real matrix spaces

I got the following problem: Let $S:\mathbb{M^R}_{3 \times 3} \to \mathbb{M^R}_{3 \times 3}$ be a linear transformation defined by $S(A) = (3A+A^T)/2$ for every matrix $A \in \mathbb{M^R}_{3 \times ...
3
votes
1answer
70 views

Is this determinant identity true?

I simulated the following $$\det(I+[A|B][A|B]^*)\geq\det(I+[B][B]^*)$$ and every time I get a true result. So how can I prove this statement? Here $[A|B]$ is matrix augmentation. $I$ is the identity ...
6
votes
3answers
343 views

Number of matrices with no repeated columns or rows

If you consider all $10$ by $15$ matrices with entries that are either $0$ or $1$, there are ${2^{15} \choose 10}$ with no repeated rows (up to row permutation) and ${2^{10} \choose 15}$ with no ...
4
votes
1answer
62 views

Show that if $tr(A+B) > tr(A)$ then $tr((A+B)^k)\geq tr(A^k)$ for any $k\geq 1$

This may be a stupid question, but I am completely stuck, I don't even know where to start. I have to show that if $tr(A+B) > tr(A)$ then $tr((A+B)^k)\geq tr(A^k)$ for any $k\geq 1$, where $A$ and ...
0
votes
0answers
36 views

Pareto distribution and matrix

I am wondering if there are any bounds are known on the eigenvalues of random matrix whose entries are with Pareto distribution? Thank you.
1
vote
1answer
51 views

Linear Algebra Span question

Let $a, b, c$ be vectors in $\mathbb{R}^3$. From what I understand, if $c\in \mathrm{Span}\{a,b\}$, then $b\in \mathrm{Span}\{a,c\}$. Since they all fall on the same plane, I can't seem to find a ...
2
votes
1answer
20 views

Factorizing a block column matrix with element-wise factors

Is it possible to factor this matrix $$\begin{bmatrix} x_{11} a_{11} & x_{11} a_{12} & x_{12} a_{11} & x_{12} a_{12} & \\ x_{21} a_{21} & x_{21} a_{22} & x_{22} a_{21} ...
0
votes
2answers
31 views

Linear Algebra Matrix Question solutions

Hi I was just wondering if an augmented matrix had no pivot positions, would the system have infinite solutions? Since it has no pivot positions that means, the columns must be filled with 0s and it ...
1
vote
2answers
128 views

Showing that $A=B+\alpha \cdot I$ is an invertible matrix

Let $B$ be a non-zero random $n\times n$ matrix generated using the matlab command $B=rand(n,n)$. I need to show that $A=B+\alpha \cdot I$ is an invertible matrix, where $\alpha=\|B\|_{\infty}$. I ...
0
votes
1answer
29 views

The entries of a random matrix

I am confused about something. Whenever I create an $n\times n$ random matrix using Matlab (using the command $A=\mathrm{rand}(n,n)$), I get a square matrix whose entries are all between $0$ and $1$. ...
0
votes
2answers
57 views

Vector space and Dual space

I'm struggling with this problem: Let $V$ be a vector space over a field $F$ and let there be $l_1,l_2 \in V^*$. I need to show that if $l_1(x)l_2(x)=0$ for every $x \in V$ then at least one of ...
0
votes
1answer
37 views

Two different ways to write C(A)?

let $\mathrm A \in \Bbb R^{m\times n}$ I know that the three fundamental subspaces are: $\mathrm \ker(\mathrm A) = \{ x \in \Bbb R^n : \mathrm Ax = 0 \} = \{x\in \Bbb R^n : \langle ...
1
vote
2answers
161 views

Combine a rotation matrix with transformation matrix in 3D (column-major style)

I'm trying to concatenate a rotation matrix with a 4x4 homogeneous transformation matrix with column-major convention. I want this rotation matrix to perform a rotation about the X axis (or YZ plane) ...
0
votes
0answers
26 views

Comparing two matrices over row and column swap operations

Is there a way to compare if two matrices are equal over all row and column swap operations faster than the obvious (try all the combinations) way?
5
votes
1answer
73 views

Eigenvalues appear when the dimension of the Prime Index Matrix is a prime-th prime. Why?

I had a look at the eigenvalues of the matrix, I called it Prime Index Matrix (is there a better name?), constructed like the following: $$ P_{k,p_k}=P_{p_k,k}=1, $$ where $p_k$ is the $k$th prime. ...
1
vote
1answer
101 views

Derivative of matrix inverse w.r.t. vector

I need to differentiate the inverse of the $K\times K$ symmetric matrix $A$ w.r.t some vector (that $A$ depends on). Is there a rule for this? In case I do the derivative w.r.t. to some scalar there's ...
0
votes
0answers
18 views

Matrix Optimal Strategy Problem

(B) What is the expected value of the game for R if the bank R always chooses TV and bank C uses its optimum strategy? E= _ (type fully reduced fraction or mixed number) (C) What is the expected ...
0
votes
1answer
44 views

Inequality matrix norm

Let $A$ be an $n\times n$ random matrix $A=rand(n,n)$. Let $\alpha=max_{i,j}|a_{ij}|$ (i.e, $\alpha$ is the largest entry in $A$ in absolute value).I need to show that $\ \alpha < \| A \|_{2}$. ...
1
vote
1answer
292 views

Need help in understanding how to find an elementary matrix

I read this chapter in my book and thought I understood it, but I don't. I tried working a problem to test my understanding and I just don't know how to get started. Given the following matrices: ...
1
vote
1answer
65 views

Getting stonewalled on computation of $2\times 2$ Hessian matrix

The question: Let $z \in R^N$, and let $f(z) = \log[1^T z] \in R$. I am told that the Hessian matrix of this function is the following: $$ H = \frac{1}{1^Tz}\Big[ 1^Tz \mathrm{diag}(z) - zz^T \Big] ...
0
votes
1answer
24 views

Understanding matrix equations

If I have two matrices $ A = \begin{bmatrix} 1 & 1 & 2 \\ 1 & 2 & 3 \\ 0 & -3 & -2 \\ \end{bmatrix} $, $ AX = \begin{bmatrix} 2 & ...
0
votes
1answer
65 views

Determinant of a symmetric, positive semidefinite, sparse integer matrix

I'm looking for an algorithm that calculates the (log) determinant of a symmetric, positive semidefinite, sparse integer matrix. Does such an algorithm exist that can exploit both sparsity and ...
-2
votes
1answer
77 views

Product of matrices: is $A\times A=0 \implies A = 0$ true or false? [closed]

Is the statement $A \times A = 0 \implies A = 0$ (where $A$ is a square matrix) true or false? If it is true, prove it. If it is false, give a counter-example. Edited I did not know about nilpotent ...
2
votes
0answers
22 views

properties of row reduction to explain $Ax - b = 0$ being true

How would i go about using properties of row reduction to explain why the equation $Ax - b = 0$ is true? I am not sure how to attack this. I know that $Ax=b$ where $b$ is a linear combination of the ...
0
votes
2answers
184 views

determine whether the equation $Ax = b$ is consistent for every $b$ in $\mathbb R^m$

I have two problems, the first one is the following matrix: $$\begin{bmatrix}1 & 0\\ -2 & 1\end{bmatrix}$$ where the RREF is $$\begin{bmatrix}1&0\\0&1\end{bmatrix}$$ and where the ...
2
votes
1answer
122 views

Linearly independent functionals

Let $ f_1,\ldots,f_n$ be linearly independent linear functionals on a vector space $X$. Show that there are $n$ elements $x_1,\ldots,x_n$ in $X$ such that the $n\times n$ matrix $[f_i(x_j) ]$ is ...