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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3answers
51 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
22 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
0answers
44 views

Transformation matrices and hermitian/unitary/normal/… matrices

I need some help with the following - have I done the correct things or how can I solve the task? Let $f \in End(V)$, V a unitary space $\mathbb{C}^3$ given by: $A_{\alpha \beta} (f) = \frac{1}{7} ...
0
votes
1answer
111 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 ...
0
votes
1answer
62 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 ...
2
votes
2answers
347 views

How prove this matrix limit is $\lim_{m\to\infty}A^mx=\left(\dfrac{e}{n}\right)$

Question: let $A$ is Doubly stochastic matrix,and the eigenvalue such $$\lambda_{1}=1,|\lambda_{j}|<1,(j=2,3,\cdots,n)$$ and the $$e=(1,1,1,\cdots,1)^T$$ show that : for any vector ...
1
vote
1answer
62 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 = ...
-3
votes
1answer
60 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 $\bf X$, ...
7
votes
2answers
92 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
62 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.
0
votes
1answer
101 views

Important topics in Matrix analysis

I'm doing a course in Matrix analysis, and I'm supposed to prepare a presentation about any topic in Matrix theory. We already covered the book "Matrix Analysis" by Horn, so preferably I need a topic ...
10
votes
1answer
889 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 ...
0
votes
1answer
40 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 ...
0
votes
2answers
81 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 ...
2
votes
1answer
53 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 $x \in \mathbb{R}^n$, ...
1
vote
1answer
45 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 ...
1
vote
1answer
32 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 ...
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
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
52 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 ...
0
votes
1answer
51 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 $P^{-1}$ ...
4
votes
2answers
77 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 ...
3
votes
6answers
312 views

Given an $n \times n$ matrix $A$, if $Ax = x$ for all $x \in \Bbb R^n$, prove that $A$ is the identity matrix.

How can I prove that this statement is true? I found this in an old textbook I was flipping through and was wondering how I could construct a proof for it.
1
vote
0answers
11 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
2answers
122 views

How to find matrix $A$ given $Ax=b$. Also $det(A)$ & $sum(A)$ are known. [duplicate]

Here is an example: $A = \begin{bmatrix} 2 & 3 \\ 5 & 1 \end{bmatrix}$ $x = \begin{bmatrix} 3 \\ 8 \end{bmatrix}$ $b = Ax$ so $b = \begin{bmatrix} 30 \\ 23 \end{bmatrix}$ Now i want to ...
0
votes
1answer
198 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 ...
1
vote
1answer
46 views

Eigen Values and Nature of Matrix [closed]

Let J be a 3x3 matrix all of whose entries are 1.Then (i)0 and 3 are the only eigen value of J (ii)J is positive semi definite. (iii)J is diagonalizable (iv)J is positive definite. Here, J is ...
4
votes
3answers
103 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 ...
7
votes
3answers
99 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 ...
1
vote
1answer
33 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
2answers
2k views

Rotate a point around another point by an angle

In a two-dimensional matrix, how do I go about rotating a point A around a point B by Z degrees like in the problem below:
2
votes
4answers
109 views

Non-negative, real matrix $\Rightarrow$ non-negative, real eigenvalues?

Does a matrix with all non-negative, real entries have all non-negative, real eigenvalues? Where might I find a proof of such? Ideas: Perhaps we can multiply a prospective eigenvector so its biggest ...
3
votes
1answer
40 views

Comparing polar- and Cartan decomposition

Comparing polar- and Cartan-decomposition, can we conclude that every positive definite symmetric matrix $A\in\text{Gl}(n,\mathbb{R})$ can be written as $A=\text{diag}(\lambda_1,..,\lambda_n)\cdot P$ ...
0
votes
1answer
37 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
573 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
22 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{ ...
1
vote
1answer
29 views

Equation of matrices

Let $V$, a 3d vector space above $F$. Let $T:V\rightarrow V$, linear transformation and $E$, an "ordered" basis such that: $$[ T ]_E = \left( \matrix{ 0 & 0 & a \cr 1 & 0 & ...
0
votes
1answer
40 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 & ...
1
vote
1answer
202 views

Is it trivial that I will always find a solution to Laplace's equation via finite-difference method

I've followed the method explained in Numerical Recipes in C, chapter 19, to solve a elliptic equation: http://www.capca.ucalgary.ca/top/teaching/phys499+535/PHYS535/nrf90/pde-c19-0.pdf I'm ...
1
vote
1answer
47 views

Any name for a special matrix with only non-zero entry

Consider an $n\times n$ matrix $\mathbf{E}_{ij}$ which is 1 at entry $(i,j)$ and zero everywhere else. Is there any special name for this kind of matrices?
2
votes
0answers
21 views

Finding the matrix ${\left[ T \right]_E}$

Let the matrix ${\left[ T \right]_{B \to E}}$, the matrix where: $${\left[ T \right]_{B \to E}}{\left[ v \right]_E} = {\left[ {T(v)} \right]_B}$$ It's given that: $${\left[ T \right]_{B \to E}} = ...
1
vote
4answers
116 views

Proving the standard matrix U of T to be orthogonal

So my class is getting into orthogonality, however, our reading assignments haven't been touching on transformations. I have this proof problem that I cannot seem to get around. Does anyone have any ...
0
votes
1answer
60 views

Why is this proof valid - inverse function theorem

Question from worksheet, I don't fully understand the solution the teacher gave. Question: let $S$ be the set of symmetric positive definite matrices of dimension $n$x$n$. Let $T: S \to S$, ...
2
votes
1answer
53 views

How to calculate the determinant of a matrix?

There are three $2\times 2$ matrices A, B and C they satisfy the following relation \begin{equation} A_{ij}=B_{ij}+[CB+(CB)^T]_{ij}x+(CBC^T)_{ij}x^2, \end{equation} where $x$ is an arbitrary variable. ...
4
votes
6answers
228 views

Find maximal possible determinant value given constraint

Task is to find maximal possible determinant value for 2x2 and 3x3 matrices given following constraint: $$\sum_{i,j=1}^na_{ij}^2 \le 1$$ I was able to come up with solution, but I received the test ...
2
votes
1answer
111 views

Matrix Norm Inequalities and Linear Least Squares

I am working through one of my universities old QUALS and came across the following problem: Let $A,E\in \mathbb{C}^{m\times m}$. Suppose that $\sigma_\min>0$ is the smallest singular value of ...
1
vote
3answers
434 views

Find the row echelon form of a 4x4 matrix

I want to row reduce the following matrix into an echelon form: \begin{pmatrix} -(-\beta+\alpha)^{1/2} & 0 & 1 & 0\\ 0& -(-\beta+\alpha)^{1/2}& 0 & 1\\ -\beta & \alpha ...
1
vote
1answer
352 views

Hermitian Matrix Unitarily Diagonalizable

I am having trouble proving that Hermitian Matrices ($A = A^{*}$) are unitarily diagonalizable ($A = Q^{*}DQ$, where Q is a unitary matrix, $QQ^{*} = I$ and D is a diagonal matrix). I also know that ...
0
votes
0answers
112 views

norm of a nilpotent matrix

A proof I was reading used the claim that $||{N}||_2$ = 1 for a nilpotent matrix $N$. I tried to prove it, and have a couple of questions on it. First, my "proof": We know that there exists a basis ...