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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1
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0answers
42 views

element subset for adjacency matrix

I am trying to create an element of a matrix that is a subset of a larger matrix. However, I am told that my subscripts do not match. I wanted other people's opinion as to what I am doing wrong and ...
0
votes
1answer
42 views

Rank of matrice: proof

I don't understand how to prove this property: $n \in \mathbf{N}$, $A \in \mathbf{L(V)}$ with $X$ and $Y$ being two basis. Then why is $rk(A) = rk(^{X}A^Y)$ true?
3
votes
3answers
297 views

Do row and column permutations generate all permutations?

Suppose $m,n\ge 1$ are integers. Do row and column permutations of an $m\times n$ matrix generate the group of all permutations of the $mn$ entries of the matrix? More formally, let $A_1$ be the ...
0
votes
1answer
27 views

Basic matrice notation

I want to compute the L2 distance between a set of points X and M using matrices, for that I proceed as follows: 1) I substract both matrices, X-M 2) I square each matrice member (X-M)^2 3) I ...
1
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0answers
36 views

Extension of Schur-Cohn for quadratic matrix equation

Starting from a quadratic in $z\in\mathbb{C}$ with real scalar coefficients $b,c$: $z^2 + bz+c=0$ and using the Schur-Cohn recursion, I can get the following conditions on $a,b,c$ such that $|z|\leq ...
0
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2answers
192 views

Projection matrix to project a point in a plane

How to determinate the 4x4 S matrix so that the P gets projected into Q, on the XZ (Y=0) plane? Q = S P
0
votes
1answer
164 views

Finding loci of possible points satisfying vector simultaneous equations

I recently had an exam and a question came up which I was only partially able to answer. The question was the following: Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be constant vectors in ...
2
votes
0answers
114 views

Linearly independent skew symmetric complex matrices having the least eigenvalues

Question: Let $A$, $B$ be two $5 \times 5$ (or $7 \times 7$) skew-symmetric complex matrices (i.e. $A^t = -A$), and suppose that $$ \forall t,s \in \mathbb{C}, \quad M(t,s):=(tA+sB)^*(tA+sB) \text{ ...
0
votes
0answers
54 views

The determinant of a matrix

In order get the determinant of$$\begin{pmatrix} \lambda-n-1 & 1 & 2 & 2 & 1 & 1 & 1& 1 & \cdots &1 & 1 \\ 1 & \lambda-2n+4 & 1 & 2 & 2 &2 ...
1
vote
0answers
69 views

Block Diagonalization related to Direct Sum and Single Eigenvalue?

I'm just a beginner in Linear Algebra, and I've proved myself the following: A matrix $A^{n \times n}$ is block diagonalizable if and only if the base field $F^n$ can be divided into at least two ...
2
votes
1answer
208 views

Hessian matrix to establish convexity

I have a function, $u(x_1,x_2)=\alpha \ln(x_1)+(1-\alpha)\ln(x_2)$. where $0<\alpha <1$ I want to prove that it is convex. The Hessian matrix I have constructed is: $$ \left( ...
-1
votes
2answers
92 views

A question on Rank and trace of a special matrix [closed]

I want to share the following question which was asked in a competitive exam: For a fixed positive integer $n\geq 3$, let $A$ be the $n\times n$ matrix defined by $A=I-\dfrac{1}{n}J$, where $J$ is ...
2
votes
1answer
188 views

Some questions on Nilpotent matrix [closed]

Q & A style. Just wanted to share the following question which came in a competitive exam and so college level maths students may find it useful: A non-zero matrix $A\in M_n(\mathbb{R})$ is said ...
11
votes
2answers
200 views

Prove or disprove : $\det(A^k + B^k) \geq 0$

This question came from here. As the OP hasn't edited his question and I really want the answer, I'm adding my thoughts. Let $A, B$ be two real $n\times n$ matrices that commute and $\det(A + ...
0
votes
0answers
113 views

How to find a transformation matrix T?

(1.) Suppose there is matrix C of dimension "p by n" where p is less then n i.e. p I want to know is there any particular way exist to find transformation matrix T of dimension "n by n" such that ...
0
votes
2answers
1k views

How to compute the eigenvalue condition number of a matrix

How to compute the eigenvalue condition number, $\kappa(4,A)$, of a matrix $A$ $$A = \begin{bmatrix} 4 & 0 \\ 1000 & 2\end{bmatrix}$$ I am a bit stuck on how to proceed solving this problem ...
1
vote
1answer
107 views

linear recursion $y_n=A \cdot y_{n-1}$

Let $a,b, \in \mathbb{R}$. Let $x_0=a, x_1=b$ and $x_n=\frac{x_{n-1}+x_{n-2}}{2}$ for $n \geq 2$ (i) Write the recursion in the form $y_n=A \cdot y_{n-1}$ where $A$ is a $2 \times 2$ matrix and ...
1
vote
3answers
52 views

examining if a matrix is diagonizable

I was practicing some linear algebra problems and I stopped at this one: Without calculating the eigenvectors, show that the following matrix is diagonalizable and find the diagonal matrix to which ...
2
votes
0answers
23 views

Finding how many solutions there are for a system of 3 linearly independant rows with 7 variables over $\mathbb Z_{13}$

Suppose we have a linear and homogenous system over $\mathbb Z_{13}$, with 7 variables. Suppose after doing RREF on the coefficients matrix we have 3 non zero rows. How many solutions there are to ...
0
votes
1answer
218 views

Complex function and Jacobian matrix

Given some complex-differentiable function $f:\mathbb{C}\rightarrow\mathbb{C}$ defined $f(x,y)=u(x,y)+iv(x,y)$, we know the Cauchy-Riemann equations hold, so: $$\dfrac{\partial u}{\partial ...
1
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1answer
153 views

Is there an elementary proof that a positive real matrix has a positive real eigenvalue?

It's an exercise in, e.g., Bredon's "Geometry and Topology" that a square matrix with positive real entries necessarily has a positive real eigenvalue. (Sketch: consider the intersection of the ...
1
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1answer
68 views

An example on my book that asks for the basis of an eigenspace

Let $$ A = \begin{bmatrix}4&-1&6\\2&1&6\\2&-1&7\end{bmatrix}$$ An eigenvalue of A is 2. Find a basis for the corresponding eigenspace. Solution: Form $$A-2I = ...
1
vote
1answer
394 views

Trying to use Cholesky decomposition of covariance matrix to sample error ellipsoid

I'm trying to construct an error ellipsoid from a covariance matrix (which exists for a 3D point) and then sample consistent xyz points in this region. In a previous question when I asked about this ...
7
votes
2answers
104 views

The map that sends $A$ to its greatest eigenvalue is continuous.

The map $f:S_n(\mathbb R)\to \mathbb R$ such that $f(M)$ is the greatest eigenvalue of $M$ is continuous ($S_n(\mathbb R)$ is the set of symmetric matrices) I need to prove this result in order ...
0
votes
3answers
66 views

Does R' = R*R'*R^(-1) hold?

Given two 3D rotation matrices ($3\times 3$) $R$ and $R'$, does this equivalence hold?: $R' = R*R'*R^{-1}$ My intuition tells me so, but I can't find a formal proof for it. Thanks.
2
votes
3answers
214 views

Matrices and rank inequality

Let $A \in K^{m\times n}$ and $B \in K^{n \times r}$ Prove that min$\{rk(A),rk(B)\}\geq rk(AB)\geq rk(A)+rk(B)-n$ My attempt at a solution: $(1)$ $AB=(AB_1|...|AB_j|...|AB_r)$ ($B_j$ is the j-th ...
-3
votes
1answer
76 views

Eigenvalues and Eigenvectors of a singular Covariance matrix

I am working on a research in which my data matrix $\bf X$ has dimension of $N\times P$ where $P>>>>N$.ie. its a small sample size problem. I need to compute the covariance of $\bf X$, ...
0
votes
1answer
41 views

Orthogonal diagnoal Matrix

I'm having problem for orthogonal diagonalization a matrix. A=\begin{pmatrix} 1 & 1 & 0\\ 1 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} I want to know what I'm doing wrong. So in order ...
2
votes
1answer
60 views

How do I show the covariance matrix of a multivariate normal random vector is positive definite?

The question is as follows: Suppose the $n$-dimensional random vector $\textbf{Z}$ has mean vector $\mu$ and variance-covariance $V$. By considering $Var(x^{T}\textbf{Z})$ for $x \in \mathbb{R}^n$, ...
0
votes
2answers
110 views

Orthogonal matrix and eigenvalues

How can I find an orthogonal matrix that can diagonalize the next matrix: $$M = \begin{pmatrix} \ a & b \\\ b & a \end{pmatrix}, b\ne 0.$$ Another question is how can I find the eigenvalues ...
0
votes
1answer
26 views

linear algebra - eigenvalues/vectors & diagnalization

$$R(θ) = \begin{pmatrix} \cosθ & -\sinθ \\ \sinθ & \cosθ \end{pmatrix}$$ $0 < θ < π$ Now, I understand that there are not any eigenvectors/values over $\mathbb R$ (but do has over the ...
1
vote
1answer
35 views

Why is $\left( \begin{smallmatrix} x & y \\ y & t \\ \end{smallmatrix} \right)$ orthogonally similar to this?

When working on a problem, I encountered the following statement. Let $x,y,t \in \mathbb R$ $\left( \begin{smallmatrix} x & y \\ y & t \\ \end{smallmatrix} \right)$ is orthogonally ...
1
vote
1answer
46 views

linear algebra and matrices, dimension

Let $W = \{p(B) : p\ \text{is a polynomial with real coefficients}\}$, where $B = \begin{pmatrix}0 & 1 & 0\\0 & 0 & 1\\1 & 0 & 0\end{pmatrix}$. The dimension $d$ of the ...
0
votes
1answer
38 views

Prove that number of nonzero elements in inverted matrix is at least 2n

Let $A$ is invertible matrix $n\times n$ with $ a_{ij} > 0 $ for every $i,j$. Prove that number of elements that equal to zero in $A^{-1}$ is less or equal to $n^2-2n$. In other words, $A^{-1}$ ...
2
votes
0answers
53 views

$(\sigma_0^{-1} SL(2,\mathbb{Z}) \sigma) \cap SL(2, \mathbb{Z})$ is a right coset of $\Gamma_0(m)$ in $SL(2, \mathbb{Z})$

Let $\Gamma_0(m)$ be the subgroup of $SL(2,\mathbb{Z})$ which is defined as follows: $$\Gamma_0(m)=\left\{ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in SL(2, \mathbb{Z} : c ...
1
vote
1answer
49 views

Is $\lVert Ax \rVert^2 - \lVert Bx \rVert^2 = \lVert AA^T - BB^T \rVert$?

For matrices $A,B\in\mathbb{R}^{m\times n}$ and for any unit vector $x$, is the following true, and if so why? $\lVert Ax \rVert^2 - \lVert Bx \rVert^2 = \lVert AA^T - BB^T \rVert$ Equivalently, is ...
11
votes
1answer
1k views

Example of similar matrices $A$ and $B$ such that products $AB$ and $BA$ are not similar

I'm looking for the simplest possible example of square matrices $A$ and $B$ such that $A$ is similar to $B$, $AB$ is not similar to $BA$. Such an example should exist, but I would like to find ...
4
votes
2answers
94 views

A question about invertible matrices, $A,B$ are invertible matrices, $AB+BA=0$, show that n is even

Let $A,B\in M_n(\mathbb R)$ be invertible matrices, and let $AB+BA=0$, show that n is even. I know what the solution is: $AB=-BA\Rightarrow |1|=(-1)^n|1|\Rightarrow \text{n is even}$. So we ...
0
votes
1answer
71 views

Find unitary matrix so that $ P^{-1}BP$ is diagonal.

given is the matrix $ B = \begin{pmatrix} 1 & i & -i \\ -i & 2 & 0 \\ i & 0 & 2 \end{pmatrix} $. I have to find a matrix $P \in U(3)$ (in unitary group, meaning that $P^{-1}$ ...
1
vote
0answers
14 views

Symbol or name for Basismatrix of Linear Programming

This question is about the Basismatrix in the context of Linear Programming. Basically (haha!) we have the Matrix of the standard (or normal) form, which consists of (A|E) with the coefficient matrix ...
0
votes
1answer
316 views

Geometric meaning behind matrix-vector multiplication

Consider a matrix $A (m , n)$ and a vector $x (n , 1)$. I understand what the equation $Ax = b$ means ($A$ is transformation matrix and so on). I know what happens to $x$ due to this linear ...
2
votes
5answers
13k views

Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to ...
7
votes
3answers
101 views

Solving $X+X^T=tr(X)M$

Let $M$ be a $n\times n$ complex matrix. Solve the equation $X+X^T=tr(X)M$ where $X$ is a $n\times n$ complex matrix. I've done some case-checking. Suppose $X$ is a solution. if ...
4
votes
3answers
127 views

Prove or disprove that trace of matrix $X$ is zero

I was trying to solve a question from a competitive exam paper. This is a part of that question. Let $I_n$ and $O_n$ be $n\times n$ identity and null matrices respectively.Let $S$ be $2n\times ...
1
vote
1answer
37 views

how to prove the existence of a solution from an infinite linear system?

I need to prove the existence of a solution for variables $x_j$ with $j=1,2,3,\cdots,\infty$ from the linear system $$\sum_{j=1}^\infty A_{i,j}x_j=b_i (i=1,2,3,\cdots,\infty)$$ Where $A$ is a ...
4
votes
0answers
19 views

$\sqrt{X}$ where $X$ is a positive definite matrix is smooth $C^{\infty}$ [duplicate]

I'm trying to prove the following statement. Let $P_n \subset Mat_{nxn}(\mathbb R)$ be the set of all symmetric positive definite matrices with real entries of size $n$x$n$. Let $\sqrt{}:P_n \to ...
0
votes
1answer
40 views

invert or transpose

Is this correct: When finding the diagonalization of a matrix $A$ of the form $QDQ^{-1}$ then if you normalize your eigenvectors instead of having to invert $Q$, you could just take $Q^t$. Just ...
1
vote
3answers
2k views

How does a row of zeros make a free variable in linear systems of equations?

I don't understand how a row of zeros gives a free variable when solving systems of linear equations. Here's an example matrix and let us say that we're trying to solve Ax=0: $$\left[ ...
0
votes
1answer
47 views

Inverse matrix - transformation

I am finding inverse matrix $A^{-1}$ and I was given hint that I could firstly find inverse matrix to matrix B which is transformed from A. $$A=\begin{pmatrix}1 &3 & 9& 27\\3 & 3 & ...
0
votes
1answer
23 views

Show there's no ordered basis $E$ with the following conditions

Let $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$ such that: $$T\left( {\matrix{ x \cr y \cr } } \right) = \left( {\matrix{ 2 & 1 \cr 3 & 4 \cr } } \right)\left( {\matrix{ ...