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
2answers
49 views

Inverse of a Matrix(shortcut and tricks)

Can someone tell me if there is any shortcut or trick of finding the inverse of a matrix and not by elementary operations? Also is it possible to judge an inverse of a matrix by judging the options ...
1
vote
1answer
28 views

A question on Involuntary matrices [closed]

If A is a square matrix such that $A^2= I, then A^{-1} $is equal to what? (where I is the identity matrix)
6
votes
5answers
93 views

Let $A$ be a $2 \times 2$ real matrix such that $A^2 - A + (1/2)I = 0$. Prove that $A^n \to 0$ as $n \to \infty$.

Question: Let $A$ be a $2 \times 2$ matrix with real entries such that $A^2 - A + (1/2)I = 0$, where $I$ is the $2 \times 2$ identity matrix and $0$ is the $2 \times 2$ zero matrix. Prove that $A^n \...
4
votes
2answers
95 views

Matrices problem: $AB=B$ and $BA=A$, what is $A^2+B^2$?

If $A$ and $B$ are two matrices such that $AB=B$ and $BA=A$, then $A^2+B^2$ would be equal to?
2
votes
3answers
46 views

Finding the matrix of this particular quadratic form

I have been working on problems related to bilinear and quadratic forms, and I came across an introductory problem that I have been having issues with. Take $$Q(x) = x_1^2 + 2x_1x_2 - 3x_1x_3 - 9x_2^...
0
votes
0answers
35 views

Similarity of matrices with Jordan Form

My problem consists in determining whether the following matrices are similar or not: $$ A = \left(\begin{array}{cccc} -1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 4\\ -1 & 0 & 1 &...
-1
votes
0answers
19 views

Projected to the null space of the matrix?

I am the moment trying to understand how this is projected to the null space of the matrix? This snippet is from this paper : http://www.golems.org/papers/StilmanIROS07-task-constrained.pdf Could ...
2
votes
1answer
54 views

Solve for third rank linear tensor equation $C_{[ij]k}U^jU^k=A_i$

Is there a way to solve a general tensor equation of the form, written in an arbitrary frame \begin{equation} C_{[ij]k}U^jU^k=A_i, \end{equation} for a tensor field $C$ of type $(0,3)$ (the square ...
1
vote
1answer
18 views

Compute covariance matrix random walk

Consider a random walk on the square lattice $\mathbb{Z}^2$ with diagonal jumps of size $2$, i.e. the jump probabilities are $$P(X_1 = x) = \begin{cases} \frac{1}{4} & \quad \text{if } ...
1
vote
1answer
33 views

Linear application

Let $f:\mathbb{R}^3\to\mathbb{R}^3$ be a linear application and let $\{e_1,e_2,e_3\}$ the canonical basis of $\mathbb{R}^3$. We know that $\operatorname{Im} f=\langle(1,1,3), (0,1,1)\rangle$ and that ...
1
vote
0answers
41 views

Eigenvalues of antidiagonal block matrix

I have an anti-diagonal block matrix $$J=\left(\begin{array}{ccc} 0 & A \\ B & 0 \end{array} \right)$$ where $A$ is $m\times m$ and $B$ is $n \times n$. Is there a trick to calculating the ...
0
votes
2answers
65 views

Eigenvalues of a $3\times 3$ symmetric matrix [duplicate]

Given a $3\times 3$ symmetric matrix \begin{equation} M= \begin{pmatrix} A & B & C \\ B & D & E \\ C & E & F\\ \end{pmatrix}, \end{equation} how do I find the eigenvalues? ...
2
votes
3answers
63 views

What's the easiest way to prove that the following matrices are 0?

So this is the problem: Let $A= \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 &-1 \\ 0 & 0 & 0 \\ \end{bmatrix}$ and $B= \begin{bmatrix} 0 & 0 & 0 \\ 1 & 0 & ...
0
votes
0answers
6 views

Primitive = Non-negative + Irreducible + 1 Positive element on main diagonal

Can anyone provide me with the proof for the sufficient condition for a matrix to be primitive as described by the definition from planetmath.org? (http://planetmath.org/primitivematrix)
1
vote
2answers
101 views

Given an $n \times n$ matrix B, is it always possible to find $n \times n$ matrices A and C, such that ABC “sums” arbitrary entries of B?

For example, let $$B= \begin{pmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \\ \end{pmatrix} $$ and let $\oplus$ be a function ...
1
vote
1answer
117 views

Show that the complex $N \times N$ matrix has the eigenvalues $\{1,2,\dots,N\}$

Show that the matrix $A \in \mathbb{C}^{N\times N}$ $$A_{nm}=\begin{cases} \frac{N+1}{2} & n=m \\ \frac{1}{e^{2 \pi i\frac{(m-n)}{N}}-1} & n \neq m \end{cases}$$ has the eigenvalues $...
5
votes
1answer
90 views

Is there a mathematical property which could help “sum up” information from certain matrix areas?

I have a matrix $$A= \begin{pmatrix} 2 & -1 & 4 \\ -3 & 8 & -5\\ 12 & -7 & 16 \end{pmatrix} $$ and I would like to create the matrix $$B= \begin{pmatrix} 6 & 5 & 6 \...
1
vote
3answers
118 views

Is A+B digonalizable if they share the same basis of eigenvectors

I was given the following statement: I know that the sum of two diagonalizable matrices is not allways diagonalizable, but i'm not sure how the added element of the shared base contributes.. I ...
0
votes
1answer
15 views

Matrices- Word problem

Three card players make an agreement which is that after each game the loser needs to double the money that the other two players have laid on the table. Each player lays at the beginning of the game ...
2
votes
3answers
36 views

Given an invertible $m \times m$ matrix $A$, is there a way to find the inverse of $A'$ where $A'$ is an infinitesimal perturbation of $A$?

Given a $m\times m$ matrix $A$ with a known $A^{-1}$ is there a simple way to find $A'^{-1}$ where $A'$ is a $m\times m$ matrix infinitesimally different from $A$?
0
votes
1answer
39 views

What is a non diagonal matrix?

Is it something like the diagonal elements are zeroes and the off diagonal elements may or may not be zeroes?
1
vote
1answer
42 views

Eigenvalues of a linear transformation of $M_{22}$

My question regards this problem: Let $T\colon M_{22} \to M_{22}$ be defined by $$T\left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right)= \begin{bmatrix} 2c & a+c \\ ...
3
votes
2answers
64 views

Computing the matrix of $Tp(x) = p'(x) + x^2 p''(x)$ relative to the basis $\{1, x, x^2\}$

Show that the operator $T \colon P_2(\mathbb{R}) \to P_2(\mathbb{R})$ given by $$ Tp(x) = p'(x) + x^2 p''(x) $$ is a linear operator. Compute the matrix $[T]_{B,B}$ of $T$ relative to the ...
1
vote
3answers
51 views

Condition where orthogonal rows imply orthogonal columns.

Given a square matrix with orthogonal, non-zero rows (that are not orthonormal); must the rows each have exactly and only one non-zero element in order for the columns of the matrix to be orthogonal ...
1
vote
1answer
23 views

Geodesics on $SO(n)$

I'm trying to prove the following exercise about the ortogonal symmetric group $SO(n)$. I have been able to prove the first two sections of the exercise but I got stuck on the third. I don't ...
2
votes
1answer
66 views

If $\mathrm{rank}(A)>\mathrm{rank}(B)$ for matrices $A$ and $B$ of size $3$ then $\mathrm{rank}(A^2)\ge \mathrm{rank}(B^2)$

Let $A, B \in M_{3}(\mathbb{R})$ two matrices so that $\mathrm{rank}(A) \gt \mathrm{rank}(B)$. Prove $\mathrm{rank}(A^2) \ge \mathrm{rank}(B^2)$ Suppose $B \ne 0_3$, otherwise it's obvious. Of ...
3
votes
2answers
107 views

why is column space on the vertical in a matrix?

Why is the "column space" on the vertical in a matrix? In my mind the column space is that space that the vectors in the matrix have created. I mean, for example take the equations: ...
0
votes
2answers
33 views

How to put derivative of composition in Jacobian matrix?

Here are two functions: $f\left(u,v\right)=u^{2}+3v^{2}$ $g\left(x,y\right)=\begin{pmatrix} e^{x}\cos y \\ e^{x}\sin y \end{pmatrix} $ I need to make Jacobian matrix of $f\circ g$. I found ...
0
votes
0answers
39 views

Computing rank of a a DFT-like matrix

Define $$\mathbf{a}_N\left(\theta\right) = \frac{1}{\sqrt{N}}\left[1, e^{-j \pi \sin(\theta)}, \ldots, e^{-j (N-1)\pi\sin(\theta)}\right]^{*} \:,$$ where $(\cdot)^*$ is transpose conjugate operation, ...
0
votes
1answer
38 views

Definiteness of square of a positive definite matrix

If $A$ is positive definite ,($\mathbf x^ \mathbf H A \mathbf x> 0$) then can we say that $A^2 $ is also positive definite?
8
votes
1answer
108 views

Find solutions near $I$ of $P^2=I$

The questions is: Using the exponential, find all solutions near $I$ of the equation $P^2=I$. I did some try with the exponential: $$e^A=I+\frac{A}{1!}+\frac{A^2}{2!}+\cdots$$ $e^A=I$ while $A=...
2
votes
1answer
39 views

Trace of Matrix Exponential closed form expression

I would like to compute the $\mathbf{tr}(e^A)$ where $A$ is some square matrix with entries that have only values of either $0$ or $1$, and $\mathbf{tr}$ is the trace operator. Are there closed form ...
1
vote
2answers
54 views

If $U^*DU=D=V^*DV$ for diagonal $D$, is $U^*DV$ diagonal too?

All the matrices mentioned are complex $n\times n$ matrices. Let $U, V$ be unitary matrices such that $U^*DU=V^*DV=D$ for a diagonal matrix $D$ with nonnegative diagonal entries. Does this imply that $...
1
vote
1answer
19 views

Composition of function

I have these two functions and I have to do Jacobi matrix of their composition $h\circ c$. $h\left(r,\phi\right)=\begin{pmatrix} rcos\phi \\ rsin\phi \\ r \end{pmatrix} $ $c\left(t\right)=\begin{...
0
votes
0answers
17 views

Stationary distribution of finite-state Markov chain in terms of determinants/products of eigenvalues

I have an $M$-state continuous-time Markov chain with transition-rate matrix $K$ (the column sums are zero), which has $M$ distinct eigenvalues $\lambda_i$, $i=1,\dots,M$. $\lambda_M=0$, so $K$ has ...
1
vote
1answer
22 views

Transformation of fourth rank tensor and its matrix form

I would like to calculate transformation of fourth rank tensor, $$ C_{ijkl}=\Sigma_{m=1}^{3}\Sigma_{n=1}^{3}\Sigma_{p=1}^{3}\Sigma_{q=1}^{3}a_{im}a_{jn}a_{kp}a_{lq}C_{mnpq} $$ where $a_{xy}$ is ...
1
vote
2answers
39 views

How to Solve a linear matrix equation of an array $M = BMC$ where $ B$ and $C$ are known

Adding to the question's description : I am doing Feature extraction from videos and i am trying to implement this one line of mathematical equation to matlab or even any algorithm . let's say I ...
1
vote
2answers
46 views

interpreting the effect of transpose in the normal equations

I have a question about the normal equation. $A$ an $m\times n$ matrix with trivial nullspace, $y$ a vector outside the range of $A$. The vector $x$ that minimizes $|| Ax - y ||^2$ is the solution to $...
3
votes
1answer
133 views

Distance from point to subset of real binominals

Prove that in $$\mathbb{R}_n[x]$$ with dot product defined as $$\langle P,Q \rangle =\int_0^1 P(x)Q(x) \, dx$$ distance from $$x^n$$ to $$\operatorname{span}(1,x,\ldots,x^{n-1})$$ is equal to $$\left({...
2
votes
1answer
41 views

Proving that $X^TX+\lambda A$ is invertible when $\lambda > 0$ ($A$ is a large matrix, see inside)

I want to show that $X^TX + \lambda \begin{pmatrix} 0 & 0 & 0 & 0 & \dots & 0 \\ 0 & 1 & 0 & 0 & \dots & 0\\ 0 & 0 & 1 & 0 & \dots & 0 \\ 0 &...
-1
votes
1answer
42 views

Proof for functions of matrix [closed]

Let $A \in \text{Mat} (n,n,\mathbb{C})$. Let $I$ be a subset of $\mathbb{R}$ or $\mathbb{C}$. Further, let $f:I\to\mathbb{C}$ and $g:I\to\mathbb{C}$ be two functions for which $f(A)$ and $g(A)$ are ...
-2
votes
2answers
29 views

How to find a unique solution, infinite solution and no solution for this matrix. [closed]

The question on my page is For what value(s) of k does the system have, no solutions, a unique solution, and infinitely many solutions? All help is appreciated! Thanks in advance
0
votes
0answers
32 views

Matrices over integer fields to solve complex polynomials.

Inspired by the fruitful answer to this question regarding numerically solving polynomial equations in terms of simpler fields (in that case representing real numbers as fractions of integers), I ...
1
vote
0answers
33 views

Notation of a function that Maps two sets into a Matrix

Given two sets $P, V$ a function $f(t)$ takes any element that belongs to $ P $ or $ V $ e.g. $ t \in P \cup V$ returns a matrix of $ 2 $ columns and $K$ rows. What is the proper notation to express ...
-4
votes
2answers
32 views

prove that the product of a vector $\vec a$ and the transpose of a vector $\vec b$ is a $n \times n$ matrix with rank $1$

I need your help in solving this question. Given two vectors $\vec a$ and$\vec b$, prove that: The product of a vector $\vec a$ and the transpose of a vector $\vec b$ is a $n \times n$ ...
7
votes
1answer
134 views

We have matrix $A\in M_{n-1\times n}(\mathbb Z)$ so that the sum of entries in each row is zero. Prove that $\det(AA^T)=nk^2.$

Problem: We have matrix $A\in M_{n-1\times n}(\mathbb Z)$ so that the sum of elements in each row is zero. Prove that $\det(AA^T)=nk^2$, where $k\in \mathbb Z$. What have I considered so far: First ...
4
votes
2answers
68 views

For which classes of matrix can the matrix exponential be easily computed?

We have diagonal matrices $A = \mbox{diag} (\lambda_1, \ldots, \lambda_n)$ for which matrix exponential has simple form $e^A = \mbox{diag} (e^{\lambda_1}, \ldots, e^{\lambda_n})$, and it can be ...
0
votes
2answers
43 views

Gram–Schmidt process

I got to use the Gram–Schmidt process on this set: $\left\{ \begin{bmatrix}0&1&0\\-1&0&0\\0&0&0 \end{bmatrix},\begin{bmatrix}0&0&0\\0&0&1\\0&-1&0 \end{...
3
votes
1answer
69 views

find two matrices A and B such that $A^2 -BA-AB+B^2 = O_{2\times 2}$

Can someone please explain this question to me Question : Construct a $2\times 2$ matrix A and B with A different from B and neither A = $O_{2\times 2}$ nor $B = O_{2\times 2}$ such that $A^2 - BA - ...