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
14 views

Can we represent the curl as a multiplication by skew-symmetric matrix?

Considering that two vectors $A \times B$ = $\hat A* B$, where $\hat A$ is a skew symmetric matrix containing elements of $A$ Can we then write the curl $\nabla \times A$ as $\partial \vec r *A$ ...
0
votes
1answer
15 views

Cholesky factorization exist?

Is there a theorem or a way to show that if I have a real and symmetric positive definite matrix $A$ and its Cholesky factorization is $A = LL^T$ then $B = L^TL$ is also positive definite? Or in other ...
0
votes
0answers
12 views

The $2$-norm of a Hermitian matrix does not exceed its $1$-norm

How to prove that the $2$-norm of a Hermitian matrix does not exceed its $1$-norm? In wiki, I see $2$-norm of matrix $A$ is $\le \sqrt{\|A\|_1\|A\|_\infty}$. But I don't know how to prove that ...
1
vote
1answer
15 views

Prove that $\det(A) \cdot v \, A^{-1} = \det(A+uv) \cdot v \, (A+uv)^{-1}$.

Let $A$ be a $n \times n$ matrix, $u$ a $n \times 1$ matrix and $v$ a $1 \times n$ matrix. If $A$ and $(A+uv)$ are invertible, prove that $$ \det(A) \cdot v \, A^{-1} = \det(A+uv) \cdot v \, ...
1
vote
1answer
609 views

Frobenius Inequality Rank

I was looking for an answer for this problem in terms of matrices, but I really don't know how to prove this result. The proposition says that: Let $A\in M_{m\times k}(\mathbb{C})$, $B\in M_{k\times ...
0
votes
1answer
15 views

Determinant of a matrix with specific main diagonal

Determine the determinant of the following matrix: $$A = \begin{pmatrix}1+a_1 &1 &\cdots &1 \\ 1 &1 +a_2&\ddots& \vdots \\ \vdots & \ddots &\ddots&1 \\ 1 & ...
0
votes
0answers
17 views

Spectral Radius of a Block Matrix

I have real matrix $P$ obtained from numerical solution (FEM) of a physical problem, as \begin{equation} P=P_1+P_2= \begin{bmatrix} A_{2n \times 2n}&B_{2n \times n}\\C_{n \times 2n}&D_{n ...
4
votes
3answers
87 views

Show that if $AA^t = A^tA$, then $A=A^t$

Suppose $A$ is a matrix with non-negative real entries. If $A^tA = AA^t$, show that $A=A^t$. My proof says: $AA^t = A^tA = (AA^t)^t$. I can't seem to get to the point of $A=A^t$ Edit: What if $A$ is ...
-2
votes
1answer
35 views

How to prove the equality of two matrix expressions

I am new to linear algebra and my question maybe too simple. I have a n-by-m matrix $D$ that its columns have unit L2 norm. Let $D_a$ be a sub-matrix of $D$ composed by some columns of $D$. I need to ...
0
votes
1answer
18 views

How to prove that the column sum for a markov matrix is 1?

As is the topic, it is obvious and easy to explain in non-math language but how do I mathematically prove it?
2
votes
1answer
49 views

Largest eigenvalues of AA' and A'A

Prove that for every real matrix $A$, the largest eigenvalue of $A'A$ equals the largest eigenvalue of $AA'$ (where ' means transpose). Thanks!
3
votes
0answers
403 views

Geometric interpretations of matrix inverses

Let $A$ be an invertible $n \times n$ matrix. Suppose we interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through ...
-1
votes
0answers
15 views

Diagonal matrices in ${\rm SL}(2, \mathbb{K})$

How can one using first order logic distinguish diagonal matrices in ${\rm PSL}(2, \mathbb{K})$ in some basis? I'm trying to do it as follows: Distinguishing maximal number of commuting ...
1
vote
1answer
196 views

Proof on the inequality involving matrix splitting and trace operator

Suppose positive definite matrices $V, B, D\in\mathbb{R}^{n\times n}$ are given, where $D$ only contains diagonal entries of $V$, i.e., $D=diag(V)$, and $X, G\in\mathbb{R}^{n\times 2}$. Could the ...
0
votes
2answers
31 views

Does there exist a steady state vector of this Markov Matrix?

Does there exist a steady state vector of Markov Matrix $$P=\begin{bmatrix} \frac{1}{2} & \frac{1}{3}\\ \frac{1}{2} & \frac{2}{3} \end{bmatrix}$$ Initially I was not sure whether to answer ...
2
votes
1answer
1k views

Lower bound on norm of product of two matrices

Let $\vert \vert . \vert \vert$ be the 2-norm. Since this norm is submultiplicative, we know that for any two square matrices $A, B \in \mathbb{R}^{n \times n}$, $$ \vert \vert A B \vert \vert \leq ...
1
vote
1answer
35 views

How to find a eigenvector with a repeated eigenvalue?

The eigenvalues of my matrix are $x_1= 1$ and $x_2=3$ I get an eigenvector $V = t~[ 4~~~~~~ 3 ~~~~~1 ]^T $ but how can I diagonalize the matrix if I have the same column repeated twice. Should I ...
0
votes
1answer
40 views

What is the determinant of the sum of a diagonal matrix and a matrix of ones?

Given a square matrix, all elements outside of the main diagonal being equal to $1,$ what is its determinant?
-1
votes
0answers
18 views

Shortest distance and Cross Product [on hold]

Show that the shortest distance from a point P to the line through Po with direction vector d is $$ ||P_oP \times d||/||d||$$. I need help writing the proof for this. So far I have: let $ ...
-1
votes
3answers
57 views

If a matrix A square is 0, does it follow that A = 0? [duplicate]

Let A be a square matrix. If $A^2 = 0$, then it follows that $A = 0$. Is there a counterexample for this? If there isn't, what kinds of explanation can I make to justify this statement?
1
vote
3answers
87 views

Let $A$ be an $n\times n$ matrix that $A^n = 0$ but $A^{n-1}\not =0$, prove that…

How would I do this problem? I think I know the definitions but don't know anything past the first step. On a side note, is there any way to improve on doing these things as i'm so lost on every ...
1
vote
0answers
7 views

Transitivity of a Boolean Matrix?

I'm wondering if there's an easy way of visually telling if a boolean matrix has transitivity? The question in particular is: ...
0
votes
0answers
3 views

Link between the cofactors of two related symmetric positive-definite matrices

Let $S = \left( s_{ij} \right)_{i,j\in\left\{1...d\right\}^2}$ be a symmetric positive-definite matrix. Let $\Sigma = \left( \sigma_{ij} \right)_{i,j\in\left\{1...d\right\}^2}$ be a symmetric ...
-4
votes
1answer
22 views

Prove whether the statement is true or sometimes false. [on hold]

Prove whether the statement is true or sometimes false. If matrix A has row of zeros, does adj(A) have it also?
0
votes
0answers
10 views

there is a unitary $U \in {M_m}$ such that $X = YU$. Why $X$ and $Y$ have the same range?

Let $X,Y \in {M_{n \times m}}$ have orthonormal column. Also there is a unitary $U \in {M_m}$ such that $X = YU$. Why $X$ and $Y$ have the same range(column space) ?
0
votes
0answers
12 views

Triangulation of matrices

Suppose that $A$ is some triangularizable matrix in $M_n(\mathbb R)$. The usual approach I know of to find a triangular matrix similar to it is to find bases for all the eigenspaces, then find their ...
0
votes
0answers
19 views

Find a matrix $P$ for a square matrix $B$ with all entries $(B)_{ij} = b$, $b \in R$. $P$ is a matrix that orthogonally diagonalize matrix $B$.

The condition must meet $$D = (P^{T})BP $$ or $$D=(P^{-1})BP$$ I'm having trouble finding a pattern for all entries and infinite square size matrix. I found a matrix P that is 5x5 with all ...
0
votes
0answers
7 views

Fast modular trace of matrix exponentiation using Fermat's little theorem for matrix

The question might be related to http://stackoverflow.com/questions/12268516/matrix-exponentiation-using-fermats-theorem but is slightly different as I only concentrate on the trace of the matrix. I ...
0
votes
1answer
331 views

Simple Probability Matrix

Consider a simple model that predicts whether you pass you next test or not based on the result of your previous test. If you pass your previous test, then you have 0.2 chance you will pass your ...
0
votes
0answers
22 views

Solution of a general linear system of equations: 4-term n-equations

I have the following system of equations.... $$y_1 = c_{11} \cdot x_{11} + c_{12} \cdot x_{12} + c_{13} \cdot x_{13} + c_{14} \cdot x_{14}$$ $$y_2 = c_{21} \cdot x_{21} + c_{22} \cdot x_{22} + ...
-1
votes
0answers
13 views

inequalities for $tr(AB)$ , where A and B symmetry, positively definite matrix

Let $A$ and $B$ be two symmetry, positively definite $n\times n$ matrix with positive eigenvalue $a_1,...,a_n$ and $b_1,...,b_n$ respectively. What's the relationship between them and $tr(AB)$? Are ...
0
votes
1answer
11 views

$X$ and $Y$ have the same range(column space). Why there is a unitary $U \in {M_m}$ such that $X = YU$?

Let $X,Y \in {M_{n*m}}$ have orthonormal column. Also $X$ and $Y$ have the same range(column space). Why there is a unitary $U \in {M_m}$ such that $X = YU$?
0
votes
1answer
34 views

What is known about optimization of spectral properties of matrices over finite fields?

[I am solving the characteristic polynomial over complex numbers but since the matrices are symmetric all eigenvalues are real] Like for symmetric $d-$regular matrices over 0/1 or 0/1/-1 what are ...
-1
votes
0answers
18 views

Derivative of a matrix function by a matrix

How can I obtain the derivative of a matrix function $f(X)=X^TX$ by matrix $X$? Does the derivative organized in matrix form? Thanks in advance.
0
votes
2answers
40 views

Dot and Cross Product Proof: $u \times (v \times w) = ( u \cdot w)v - (u \cdot v)w$

How do you prove that: $u \times (v \times w) = ( u \cdot w)v - (u \cdot v)w$ ? The textbook says as a hint to "first do it for $u=i,j$ and $k$; then write $u-xi+yj+zk$ but I am not sure what that ...
3
votes
2answers
31 views

Unitary Matrices and the Hermitian Adjoint

I saw in a definition for unitary matrices, that for a complex matrix being unitary if $M: \mathbb{C}^{n} \rightarrow \mathbb{C}^{n}$ is unitary, or: $\langle Mv, Mw \rangle = \langle v,w \rangle$ ...
2
votes
1answer
39 views

Question about matrices?

I have been learning about matrices in my math class and I am confused as to how exactly they work. Take this example: $\left(\begin{array}{c c c c c | c} 1 & 4 & 1 & 0 & 0 & ...
-2
votes
1answer
26 views

What would be the basic solution of this maximization problem? [on hold]

Maximize $P=40x_1+50x_2$ Subject to $x_1+6x_2 \leq 72$ $x_1+3x_2 \leq45$ $x_1, x_2 \geq0$
2
votes
2answers
67 views

How do you find the determinant of this $(n-1)\times (n-1)$ matrix?

It's for a proof of Cayley's Formula, I know I'm being dumb and can't see it, how do I find the determinant of this $(n-1)\times (n-1)$ matrix where the diagonal entries are $n-1$ and the off diagonal ...
1
vote
1answer
18 views

A question about unitary block matrix

For $n,m \in \mathbb N$, let $M_{n,m}(\mathbb C)$ denote the set of complex $n \times m$ matrices and put $M_{n}(\mathbb C):=M_{n,n}(\mathbb C)$. For matrices $A \in M_{n}(\mathbb C), B \in ...
2
votes
0answers
45 views

Restoring Bidiagonality to a Matrix in SVD Algorithms

Good Afternoon, I am implementing the Golub-Reinsch SVD algorithm and am having difficulty with a boundary case Given a bidiagonal matrix of the form: $$ \begin{bmatrix} b11 & ...
2
votes
2answers
50 views

What's the easiest way to find all $\alpha\in\mathbb{R}$ such that $\tiny\left(\begin{matrix}1&2\\2&\alpha\end{matrix}\right)$ is positive definite?

For which $\alpha\in\mathbb{R}$ is $$C:=\left(\begin{matrix}1&2\\2&\alpha\end{matrix}\right)$$ positive definite, positive semidefinite or indefinite? It seems to be a simple task, but for ...
6
votes
4answers
85 views

For matrices, if $AB=BA$, then does it follow that $B^{2}A=AB^{2}$?

Suppose $AB=BA$ ($A, B$ are $n\times n$ matrices). Does that mean $B^{2}A=AB^{2}$ ? I looked for counter cases and couldn't find any. I tried to prove this by multiplying both sides and comparing, but ...
1
vote
1answer
10 views

diagonalizing a matrix with random elements

Consider the matrix $A = \begin{pmatrix} cY & 0 \\ 2 & 1\end{pmatrix}$, where $c \in \mathbb{R}$ and $Y$ is a random variable that is uniformly distributed over $[0,1]$ (That is, $Y \sim ...
1
vote
1answer
43 views

Finding Matrix of Linear Transformation from $R^2 \rightarrow R^2$

Let $T: R^2 \rightarrow R^2$ be given by: $$T(x_1,x_2) = (4x_1 -2x_2, 2x_1 +x_2)$$ And let $$B = \{(1,1), (-1,0)\}$$ be a basis for $R^2$. First, I write down the matrix of $$T = ...
0
votes
0answers
12 views

relation of dim kers of AB and B operators

I try to prove For any matrixes $A_{ms},B_{sn}$ $$\operatorname{rank}{A}+\operatorname{rank}{B}-s\leq\operatorname{rank}{AB}$$ First, as for any $X$ that $BX=0$ also $ABX=0$, that ...
1
vote
3answers
23 views

Trying to figure out formula for deciding how to write Linear Transformation as a matrix relative to a basis

In these lecture notes: http://www.math.rice.edu/~hassett/teaching/221fall05/linalg5.pdf the formula (last line on first page) for finding a matrix relative to bases $B'$ and $B$ is: (1) $$ C_{B'}T ...
1
vote
0answers
9 views

Discrete fractional fourier transform matrix

I am trying to write a matlab code for some calculations based on Discrete fractional fourier transform. in this article: Optimal filtering in fractional Fourier. after equation (7) a notation Fa is ...
-1
votes
1answer
30 views

Find the non trivial solution to a matrix containing a complex number.

$$ A = [B\mid b] = \left[ \begin{array}{cc|c} -3+i & -5 & 0 \\ 2 & 3+i & 0 \end{array} \right] $$ Find the non trivial solution to A. (Solving $B x = b$) I fully understand the ...
0
votes
1answer
33 views

What are the names of these variations on the transpose of a matrix and symmetric matrices?

Is there a name for the operator that reflects a matrix over the diagonal running from the top-right to the bottom-left? For the moment, define this reflection of a matrix $A$ as $A^*$. Is there a ...