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), ...

learn more… | top users | synonyms (2)

0
votes
0answers
6 views

L2 norm of an inverse of a sum of matrices

I am trying to take the L2 norm of the following expression: $-(H^{-1} + bI)^{-1}v$, where $H$ is a psd matrix, b is a scalar, and $v$ is a vector. In particular I am having trouble with the first ...
1
vote
1answer
31 views

show that a matrix is invertible

Let $A$ be an $n \times n$ matrix such that $|a_{ii}|>\sum_{j=1,j\neq i}^n|a_{ij}|$ for each $i$. Show that $A$ is invertible. $(complex matrix) The straight forward way is to show that the ...
1
vote
0answers
35 views

matrix with fractional exponent, not getting expected output in Matlab/Octave

I have a matrix exponential function that is called a number of times in an integration routine from the heat conduction model I'm trying to implement. It works, and my results match the samples in ...
1
vote
1answer
27 views

Conditions to preserve Laplacian matrix

Let $L$ be a Laplacian matrix, i.e., $L=L^T$, $L\geq 0$ and $L1_n=0$, where $1_n$ denotes the $n$-dimensional vector with all entries equal to $1$. Now I have the transformation $\bar L=WLW^T$, where ...
2
votes
2answers
60 views

If $A$ is an $m\times n$ matrix, $B$ is an $n\times m$ matrix and $n<m$, then $AB$ is not invertible.

The question was given in the early chapters of Linear Algebra by Hoffman & Kunze, so I am trying to give a proof with only the tools given to me so far - which are mainly row reduction and ...
0
votes
1answer
5 views

Proving that $L_{22}L_{22}^T=S$ is the Schur complement of a cholesky factorization

Let $A$ be an $n+m \times n+m$ symmetric positive definite matrix. $A=\begin{bmatrix}A_{11} & A_{12}\\ A_{12}^T & A_{22}\end{bmatrix}$ where $A_{11}$ is an $n \times n$ matrix, $A_{12}$ is an ...
0
votes
1answer
355 views

Principal axis of a matrix

I try to find the definition of the main axis of a matrix. I saw this phrase in some exercise: Let $A$ be a positive matrix, $f:G\longrightarrow \mathbb{R}$ a smooth function, $G$ an open set in ...
0
votes
1answer
15 views

Given a parametric solution $\vec{x}(t)$ to $Ax = b$, how can I choose the parameter $t$ so that all entries in $\vec{x}(t)$ is between 0 and 1?

Given a solution to the matrix equation $A\vec{x} = \vec{b}$ on the form $\vec{x}(t)$, how can I choose the parameter t such that all entries in $\vec{x}$ are squeezed between 0 and 1? That is, for ...
0
votes
1answer
80 views
+50

Maximization of sum of functions

Let $w,a\in R^n$, and $B\in R^{n\times n}_{++}$ (the set of $n\times n$ positive definite matrices). We know that the following function (which is a specific form of the Rayleigh quotient) has a ...
0
votes
0answers
21 views

Decomposing a matrix as the product of rotations

I'm reading an article about joint diagonalization algorithms. The article states without proof that any nonsingular $n \times n$ matrix $Q$ can be decomposed as \begin{align*} Q = \prod_{1 \leq p ...
0
votes
0answers
54 views

How to solve series of 8 equations with 8 unknowns?

In this article http://www.fmwconcepts.com/imagemagick/bilinearwarp/FourCornerImageWarp2.pdf they speak of solving for a0,a1,a2,a3,b0,b1,b2,b3 but I want to know ...
0
votes
1answer
14 views

How to calculate projection matrix for quadrilateral transform?

I have a square and its 4 corner coordinates $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)$. And I have a quadrilateral with corner coordinates $(x_1',y_1'),(x_2',y_2'),(x_3',y_3'),(x_4',y_4')$ where ...
2
votes
2answers
341 views

Derivative of Hadamard product

What is the derivative of Hadamard product of two matrices with respect to one of them? I.e. what is $D(AB)$ with respect to $A$?
2
votes
2answers
36 views

$rk(A^2)=rk(B^2) \implies rk(A)=rk(B)$ is it true?

The original statement is this: given A,B matrices $n \times n$, if $A^2$ is "Left-Right equivalent" to $B^2$ then A is LR equivalent to B (is it true or false?) I know that A is LR equivalent to B ...
0
votes
1answer
41 views

Find the minimum of a function for only positive values of the vector variable

Let variable vector $\vec{q}$ of size $m\times1$, and its diagonal counterpart $m\times m$ matrix $Q=diag(\vec{q})$, for some $m\in\mathbb{N}$. Define fixed parameter $n\times1$ vectors $\vec{p}, ...
1
vote
0answers
29 views

Matrix integration by parts

It seems to me that the integration by parts rule carries over simply to the matrix case. This can be seen by applying: $(AB)' = A'B + AB'$ and then integrating for square (time dependent) complex ...
0
votes
1answer
42 views

looks like Vandermonde determinant [duplicate]

Calculate the determinant of $M = \left( {\begin{array}{*{20}c} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2 \\ \end{array}} \right)\;$. How can one calculate this? Is there a ...
0
votes
2answers
381 views

Rank of a Matrix and Echelon Form to determine ranks.

What is the meaning rank of a matrix in terms of vectors, and how does Echelon form work in determining the rank of a matrix?
0
votes
1answer
39 views

What is this matrix doing

I am trying to find out what this matrix is doing. I am trying to follow the guide: ...
0
votes
0answers
26 views

how to construct system matrix A , given only eigen values [on hold]

how to construct system matrix A if only eigenvalues are given as follows 0,2 and 4 ?
1
vote
1answer
25 views

How to determine positive or negative definite of a bordered Hessian ?

I want to determine the minimization result I get using Lagrange Multiplier method is a local minimum by determining whether the Bordered Hessian is positive definite or negative definite.(Hopefully ...
-1
votes
3answers
146 views

$A$ is a $n \times n$ matrix over $\mathbb{R}$ such that $A^2+A+5I=0$. Find the characteristic polynomial of the matrix $A$. [on hold]

$A$ is a $n \times n$ matrix over $\mathbb{R}$ such that $A^2+A+5I=0$. Find the characteristic polynomial of the matrix $A$. it is a question from a test i had yesterday and this is how it was ...
0
votes
0answers
15 views

Can the transposition of an arbitrarily-sized matrix be broken up to smaller transpositions?

I'm working with binary matrices. Let's assume that I have an algorithm that is very efficient in transposing 8×8 or 8×16 matrices, but I would like to transpose matrices with an arbitrary size. ...
-2
votes
0answers
16 views

i want the answer with procedure to the below question can anyone please help with that… [on hold]

Q) In a Euler Angles body Attitude Representation (described below) of a UAV please make the following conversions: Attitude: Alpha = 30 degrees, beta = 15 degrees, Gamma = 45 degrees, where Alpha, ...
1
vote
0answers
16 views

Entries to find a positive definite matrix as solution of Lyapunov equation.

Given the Lyapunov equation below: $AX + XA^T + B = 0$ with $B=bb^T$ I just want to simulate A $\in \mathcal{M}_{p,p}(\mathbb{R})$ and b $\in \mathbb{R}^{p}$ to find X, solution definite positive. ...
0
votes
3answers
33 views

Dimensions of a basis of a coordinate space

I need a little clarification on the relationship between the basis, its dimension and their corresponding real coordinate space. Suppose we are operating in the fourth coordinate space ...
1
vote
2answers
34 views

What is the difference between orthogonal and orthonormal in terms of vectors and vector space?

I am beginner to linear algebra. I want to know detailed explanation of what is the difference between these two and geometrically how these two are interpreted?
0
votes
3answers
31 views
0
votes
1answer
29 views

Prove whether the linear equations are solvable or not?

I am beginner to linear algebra. I am confused for finding the solution for following question. There are set of linear equation(m equations and n unknown) represented in the form of matrices. ...
-3
votes
0answers
25 views

Question on matrix [on hold]

So i got this question from my lecturer,and i am so dumbfounded in clarifying this question. In a football league,the price for every win,draw and lose is RM5000,RM3000 and RM1000 respectively.A team ...
2
votes
1answer
18 views

Commutation of a partial trace with an operator

Let the partial trace $\mathrm{tr}_B$ be a mapping from an endomorphism End$\left( H_A\otimes H_B \right)$ onto an endomorphism End$\left( H_A \right)$. Then the partial trace is defined as $$ ...
20
votes
3answers
656 views

Old AMM problem

I am working on an old AMM problem: Suppose $A,B$ are $n\times n$ real symmetric matrices with $\operatorname{tr} ((A+B)^k)= \operatorname{tr}(A^k) + \operatorname{tr}(B^k) $ for every positive ...
5
votes
1answer
331 views

Rank of matrix as a difference of ranks

If $X$ is an $n \times p$ matrix of rank $r$ and $C = AX$ for some $q \times n$ matrix $A$ with rank$(A) = q$, how do I show that rank $(X(I-C^{-}C))=$ rank$(X)-$ rank$(C)$? I can show that rank ...
1
vote
1answer
52 views

Is this group of matrices a $p$-group?

Let $R$ be a discrete valuation ring with the maximal ideal $\mathfrak{m}=(\pi)$ and residue field $k$ of positive characteristic $p$. Now consider $\mathrm{M}_n(\pi^iR/\pi^{i+1}R)$, $n\times n$ ...
0
votes
3answers
78 views

Does every invertible complex matrix have an eigenvector?

Over $\mathbb{C}$ does every invertible matrix have at least one non-zero eigenvalue and an eigenvector? I'm generally confused about eigenvectors and eigenvalues. I understand that eigenvectors are ...
1
vote
1answer
71 views

Is there any direct method for Lagrange multiplier based domain decomposed problem?

In elastic problem, we often solve K * u = f, where K is the stiffness matrix, f the external force vector and u the displacement vector. I'm trying decompose the mesh to domains, using Lagrange ...
-1
votes
0answers
26 views

Signature of a complex matrix

If $A\in M_n(\Bbb R)$, we define its signature as the triple $(s^+,s^-,s^0)$, where $s^+,s^-,s^0\in\Bbb Z_{\ge0}$ denotes respectively the number of eigenvalues $>0,<0,=0$. How can we define ...
1
vote
2answers
59 views

Find Jordan Form of αA (α is a scalar, A a matrix)

In my linear algebra course I have a problem which goes as follows: Suppose A is an nxn matrix over field (R) And J(A) is the jordan form of A. Given α belongs to field R, what is the jordan form of ...
1
vote
1answer
25 views

What can be said about a matrix with a constant column of ones with entries from a finite field?

I am working with matrices of the following structure: $A = \begin{pmatrix} 1&\alpha_{21}&\cdots&\alpha_{n1}\\ 1&\alpha_{22}&\cdots&\alpha_{n2}\\ ...
0
votes
0answers
23 views

A question in matrix polynomial.

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
0
votes
1answer
34 views

What is wrong with matrix [[1,.5,0] [0,0,0] [0,.5,1]] steady state?

I know that Markov matrices have steady state since they always have eigenvalue $\lambda = 1$. We just solve the system of equations $A\vec x = 1 \cdot \vec x$ or $$\begin{cases} k_{a\to a} a + ...
1
vote
1answer
46 views

Vectors/matrices: How to show that $(v-w)(v-w)^t v = \frac{1}{2}\lVert v-w\rVert^2 (v-w)\quad$ (vector notation)

Given the following details: $v \neq w$ two vectors of $\mathbb{R}^n$ with $\lVert v\rVert = \lVert w\rVert$. Let $u = \frac{1}{\lVert v-w\rVert}(v-w)$ and $H = I - 2uu^t$. Suppose $x$ is an ...
2
votes
1answer
28 views

Perpendicularity in matrix space

Let $K$ and $Q$ be symmetric real matrices such that $K+Q$ is positive semidefinite ($\ge0$). My question is two questions: Does $KQ=0$ imply $K\ge0$ and $Q\ge0$? Does trace$(KQ)=0$ imply $K\ge0$ ...
1
vote
1answer
30 views

Why is the largest element of symmetric, positive semidefinite matrix on the diagonal?

I know the very well know equivalence of the properties of a positive, semidefinite matrix: $A$ is positive semidefinite, $A = U^T U$ for some matrix $U$, $\mathbf{x}^T A \mathbf{x}\geq 0$ for every ...
1
vote
2answers
82 views

Column and row space basis

$Let \ A = \left[\begin{array}{cccc}1&2&-1&1\\2&4&-3&0\\1&2&1&5\end{array}\right]$ Using Gauss elimination, lead matrix A to row reduced echelon form: ...
2
votes
1answer
49 views

Diagonalization and Commuting Matrices

Attempt: I have shown part (ii) and I have found the eigenvalues and eigenvectors for A, B respectively and shown they can be diagonalised. I need help with (iii) and (iv), for (iii) I can't show ...
1
vote
2answers
40 views

Finding a basis for the null space of a matrix

The problem is to find the null space of the matrix $$ A = \begin{bmatrix}1&2&1&-3\\2&4&4&-1\\3&6&7&1\end{bmatrix}. $$ Does this have something to do with linear ...
3
votes
0answers
32 views

A proof involving matrices (checking working)

Matrices $A,B$ and $C$ are all $2 \times 2$ matrices and $C=A-CB$. Assuming that $(I+B)^{-1}$ exists, prove that $C=A(I+B)^{-1}$, where $I$ is the $2 \times 2$ identity matrix. I was wondering if ...
2
votes
1answer
25 views

$U^TA_1V$ is a rank-one matrix?

To give a little bit of context, the question I am asking is related to SVD decomposition. More specifically, we are trying to prove that the best rank one approximation for $A_1$ is $\sigma_1 u_{1} ...
6
votes
2answers
152 views
+50

Solve quadric equation system

How to solve this? For given real and symetric matrices $A_1,A_2,A_3,A_4\in\mathbb{R}^{4\times4}$ find $x\in\mathbb{R}^4$ $$x^TA_1x=0$$ $$x^TA_2x=0$$ $$x^TA_3x=0$$ $$x^TA_4x=0$$