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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24 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
13 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
14 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, ...
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1answer
41 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 ...
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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. ...
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3answers
31 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 ...
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2answers
29 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
30 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: ...
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1answer
25 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. ...
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1answer
21 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 ...
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1answer
16 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 $$ ...
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0answers
14 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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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 ...
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1answer
50 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 ...
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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 ...
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3answers
138 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 ...
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1answer
32 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
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1answer
27 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
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0answers
31 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 ...
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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} ...
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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 ...
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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 ...
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2answers
35 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 ...
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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 ...
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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 = ...
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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
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3answers
73 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 ...
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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 \\ ...
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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
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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 ...
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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 ...
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4answers
85 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 ...
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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 ...
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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 ...
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2answers
42 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 ...
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1answer
21 views

Can functions within a matrix adjust its size?

I've been working my way through a proof, and without going into the full extent of the details it's come down to whether a function G() exists such that the 1 by 3 matrix: ...
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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 ...
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0answers
14 views

Condition number of positive definite matrix after rectangular orthogonal transformation on both sides

What is a lower bound on the condition number of $B A B^{T}$ (besides the trivial $\operatorname{cond}(B A B^{T}) \ge 1$) where $A$ is an $n \times n$ symmetric positive definite matrix, $B$ is a ...
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1answer
22 views

Determinant of 5x5 matrices

Let A and B be 5x5 matrices with det(-3A)=4 and det(B^-1)=2. Find the det(A), det(B) and det(AB). My answer : det(A)=-12 , det(B)=1/2 and det(AB)=-6. Wish to check my answer, thank you.
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1answer
30 views

Determinant of 3x3 matrices

Let $A$ and $B$ be $3\times3$ matrices with $\det(A)=10$ and $\det(B)=12$. Find $\det(AB)$, $\det(A^4)$, $\det(2B)$, $\det((AB)^T)$. Answers: $\det(AB)=\det(A)\det(B)=120$ , $\det(A^4)=10000$ , ...
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2answers
33 views

Regularity and invertibility of two parameterized matrices?

$$ C= \begin{bmatrix} 1+a & 2 & 3 & 4 & 5 \\ 1 & 2+a & 3 & 4 & 5 \\ 1 & 2 & 3+a & 4 & 5 \\ 1 & 2 & 3 & 4+a & 5 \\ 1 & 2 & ...
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1answer
40 views

What is $\frac{\partial x^TA^{-1}y}{\partial A}$?

I had trouble proving the following: If $ A\in \mathbb R^{n\times n}$ and $A$ is nonsingular, $x \in \mathbb R^{n\times 1}$, $y \in \mathbb R^{n\times 1}$, then $ \dfrac{\partial ...
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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}\\ ...
4
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3answers
122 views

Can we prove that matrix multiplication by its inverse is commutative? [duplicate]

We know that $AA^{-1} = I$ and $A^{-1}A = I$, but is there a proof for the commutative property here? Or is this just the definition of invertibility?
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0answers
21 views

Optimization problem involving semidefinite matrix variable that is constrained to be a tensor product

I would like to solve the following optimization problem. With scalar $R$ and nine (mutually orthogonal) $9$-dimensional column vectors $\vec v_i$ all given ($\vec v_i\!'$ is the row vector Hermitian ...
0
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0answers
11 views

What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula results in small errors in relation to the standard matrix inverse operation after each application, as shown here. From what I understand, the Sherman-Morrison identity ...
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1answer
33 views

What is the derivative of the ReLu of a Matrix with respect to a matrix

I want to compute $\frac{\partial r(ZZ^tY)}{\partial Z}$ where the ReLu function is a nonlinear operator $r(x)=max(0,x)$ and $Z \in\mathbb{R}^{n\times m}$ is a matrix. I am wondering also if the ...
2
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2answers
40 views

The $\exp \circ \log$ function acts as the identity on unipotent matrices.

I am working through the exercises in "Lie Groups, Lie Algebras, and Representations" - Hall and can't complete exercise 9 of chapter 2 using the provided hint. In chapter 2, Hall defines the ...