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
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1answer
13 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 ...
1
vote
0answers
16 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
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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
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 ...
1
vote
1answer
21 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 ...
1
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0answers
28 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
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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
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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 ?
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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. ...
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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
1answer
42 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 ...
1
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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
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2answers
33 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?
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3answers
31 views

How can I show that and $n\times{n}$ matrix of the form in the description has a determinant of zero for $n>2$?

In General, $n>2$, $a_{i,j}=a_{i,j-1}+1$ and the matrix will be of the following form: ...
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. ...
1
vote
1answer
24 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 ...
2
votes
1answer
17 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 $$ ...
1
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0answers
23 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 ...
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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
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1answer
51 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 $\mathrm{M}_n(\pi^iR/\pi^{i+1}R)$, $n\times n$ ...
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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 ...
1
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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
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
1answer
33 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 + ...
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 $K\ge0$ ...
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} ...
0
votes
0answers
21 views

Can we express any matrix as an outer product expansion?

Suppose $XY$ is an $m $ by $n$ matrix, where $X$ is a $m$ by $k$ matrix and $Y$ is a $k$ by $n$ matrix. $y_i$ are the columns of $Y$ and $x_i$ are the columns of $X$. How do we know that ...
1
vote
1answer
27 views

positiveness of product of matrix

If $A$ is a positive definite matrix, B is not sure but $tr(B)>0$ where $tr$ is trace, will $tr(AB)>0$ ? that is the trace of the product of those two matrices. B is not a diagonal matrix or ...
1
vote
2answers
37 views

Trace evaluation via complex analysis

We are given $U$, $V$ unitary matrices of size $N \times N$ whose spectral decomposition is known (in my specific problem, $N=4$, and $U$, $V$ are matrices with real coefficients but we can keep it ...
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 ...
2
votes
2answers
44 views

Finding $P$ such that $P^TAP$ is a diagonal matrix

Let $$A = \left(\begin{array}{cc} 2&3 \\ 3&4 \end{array}\right) \in M_n(\mathbb{C})$$ Find $P$ such that $P^TAP = D$ where $D$ is a diagonal matrix. So here's the solution: $$A = ...
-1
votes
1answer
37 views

Inverse of the sum of the inverse of 2 non-invertible matrices

Given that the following square matrices are non-invertible: $\bf A$, $\bf B$, and (A+B) UPDATE: Assume $\bf (A+B)$ is invertible. and given that $\bf (A+I)$, $\bf (B+I)$, and $\bf ...
0
votes
3answers
77 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
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0answers
27 views

prove that a antisymmetric and invertible matrix is congruent to another matrix

let A be an antisymmetric and invertible matrix $A \in M_{2m}(\Bbb{R})$ prove that A is congruent to $$ \begin{pmatrix} 0 & I_m \\ -I_m & 0 \\ ...
1
vote
1answer
22 views

Eigenvalues of Certain Symmetric Block Matrix

What can we say about the relation between the eigenvalues of the following block matrix with identity diagonal blocks, and the singular values of the off-diagonal blocks: \begin{equation} ...
1
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2answers
35 views

All the cases for Image and Kernel

here alpha is a real variable, and I need to find the kernel and image for all values for alpha. Attempt: I can't seem to figure out all the cases which I need to evaluate, as the last two columns ...
1
vote
1answer
19 views

Interlacing Theorem on Singular Values

Does the Cauchy's interlacing theorem hold for "singular values" of matrices too? I saw on this publication first Theorem that it does. It states that singular values of a matrix interlace the ...
2
votes
2answers
25 views

Show that any 2D vectors can be expressed in the form…

(a) Show that any 2D vector can be expressed in the form $s \begin{pmatrix} 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 7 \end{pmatrix},$ where $s$ and $t$ are real numbers. (b) Let $u$ and $v$ be ...
3
votes
4answers
86 views

Is there always a matrix $X$ such that $X^2=A$?

Is it true that for every $A\in M_{2\times 2} (\mathbb{C})$ there's an $X\in M_{2\times 2} (\mathbb{C})$ such that $X^2=A$? For the matter of fact, I don't have a clue, other than evaluating the ...
0
votes
1answer
26 views

Proving that same solution set implies row equivalence

The question I am trying to solve is for a much simpler case: Suppose $R$ and $R'$ are $2\times3$ row-reduced echelon matrices and that the system $RX=0$ and $R'X=0$ have exactly the same ...
1
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0answers
15 views

How to Construct $N$-dimensional Unitary Matrix Basis

Galitski's Exploring Quantum Mechanics says on its page 29, (There are $N^2$ linearly) independent Hermitian matrices of rank $N$. The number of independent unitary matrices is also $N^2$, since ...
1
vote
2answers
43 views

Are those matrices congruent?

$$A = \left(\begin{array}{cccc} 1&0 \\ 0&-1 \end{array}\right), B=\left(\begin{array}{cccc} 1&0 \\ 0&2 \end{array}\right), C = \left(\begin{array}{cccc} 1&0 \\ 0&4 ...
10
votes
0answers
85 views
+200

Bound on the difference of two determinants

Let $A$ and $B$ be two real, $n\times n$ matrices. Using Hadamard's inequality, it is not hard to show that $$ \left|\det A - \det B \right| \leq \|A-B\|_{2} \frac{\|A\|_{2}^n -\|B\|_{2}^n}{\|A\|_2 ...
0
votes
1answer
78 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 ...
4
votes
2answers
87 views

Determinant of symmetrical factorized matrix

Given $A, B \in \mathbb{R}^{n\times n}, t \in \mathbb{R}\setminus \{0\}$ with $b_{ij} = t^{i-j}\cdot a_{ij}$. Prove $\det(A) = \det(B)$. I first thought of induction. I can easily prove this for $n ...
36
votes
4answers
3k views

Solutions to the matrix equation $\mathbf{AB-BA=I}$ over general fields

Some days ago, I was thinking on a problem, which states that $AB-BA=I$ does not have a solution in $M_{n\times n}(\mathbb R)$ and $M_{n\times n}(\mathbb C)$. (Here $M_{n\times n}(\mathbb F)$ denotes ...