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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1
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1answer
10 views

Matrix translation by (1x2) vector

I'm having trouble figuring out how to approach this matrix translation question: Find the equation of the image line produced by translating all of the points on the line $y = 3x -1$ by the ...
0
votes
0answers
6 views

All permutation matrices that convert one Hadamard matrix into another Hadamard matrix.

Given a Hadamard matrix $H$, I know that applying row and column permutations, along with multiplying a row or a column with a -1 results in another Hadamard matrix $H^{'}$ equivalent to the first. ...
1
vote
1answer
21 views

Irreducible matrix equivalent connectedness of matrix graph?

If a matrix is irreducible, based on the following definition A matrix is reducible if there are two disjoint sets of indexes $I,J$ with $|I|=\mu$, $|J|=\nu$, $\mu+\nu=n$ such that for every ...
0
votes
1answer
17 views

How to construct orthogonal basis from a missing vectors?

I have $m$ vectors with a missing element each. $v_i=(*, a_{2i},\cdots,a_{ni})^\mathrm{T}\,\forall\, i\in\{1, \cdots, m\}.$ I would like to add the missing element $*$ to all $v_i$'s such that all ...
2
votes
1answer
28 views

Checking connectivity of adjacency matrix

What do you think is the most efficient algorithm for checking whether a graph represented by an adjacency matrix is connected? In my case I'm also given the weights of each edge. There is another ...
2
votes
2answers
18 views

Decomposing a square matrix into two non-square matrices

I have a matrix $A$ with dimensions $(mxm)$ and it is positive definite. I want to find the matrix $B$ with dimensions $(nxm), (n << m)$, which follows the following expression: $$A = B'B$$ Here ...
2
votes
1answer
127 views

How to prove these 2 matrix problems?

I'm reading a book and it gives that $\frac{\partial}{\partial A}Tr(AB)=B^T$, then it shows we can obtain $\frac{\partial}{\partial A}Tr(ABA^T)=A(B+B^T)$. But it seems we should have ...
0
votes
1answer
23 views

Getting linear combinations in linear algebra?

I failed a homework problem a few days ago. I can't figure out how they got the answers, which have been given in green as corrections. Help me figure how they got them;
0
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0answers
10 views

Natural logarithm of a square matrix without eigen-analysis

I'm trying to find a method to determine the natural logarithm of a square nonsingular matrix without using eigenvalues or eigenvectors. So far, I've only found this method: ...
2
votes
1answer
31 views

Effective way of checking if all eigenvalues of a matrix are integers

Given A matrix with integer entries, it should be checked if all its eigenvalues are integers. Of course, the characteristic polynomial could be calculated, but is there any faster (or easier) ...
2
votes
0answers
25 views

Which n-tuples of positive integers can be the eigenvalues of some matrix with positive integer entries?

This question is closely related to some questions I already asked Given a tuple of positive integers (such as (1,2,5) ), is there a matrix A with positive integer entries such that the integers in ...
0
votes
1answer
36 views

Invertible Matrices Proof

Given that B is an invertible matrix and $B^3 + B^4 + B^7 = I$, find an expression for $B^{-1}$ in terms of only $B$. (where $I$ is an identity matrix) $B$ is a matrix that is $n \times n$.
2
votes
2answers
23 views

2x2 inverse of a complex matrix with complex determinant

Firstly, my question may be related to a similar question here: Are complex determinants for matrices possible and if so, how can they be interpreted? I am using: $$ \left(\begin{array}{cc} a&b\\ ...
0
votes
0answers
25 views

Matrices with functions as entries

I am interested is studying matrices which have functional entries. Specifically I am looking at quadratic forms of the type $x^T Q(x) x$ where $Q(x)$ is a matrix whose entries are functions of $x$. I ...
3
votes
4answers
80 views

Can you use row and column operations interchangeably?

Is it possible to use row and column operations "at the same time" on a matrix $A$? So, for example, first subtracting $row_1$ from $row_2$, and then choosing to multiply $column_3$ by a constant $c$? ...
0
votes
1answer
37 views

If $A\ne 0$ is a square matrix over a commutative ring with $\det A=0$, then its null space contains an element whose components are minors of $A$

Let $R$ denote a commutative ring and $A\ne 0$ a $n\times n$ matrix over $R$ with $\det A=0$. Then there exists a $x\in\ker A\setminus\left\{0\right\}$ such that all components of $x$ are minors of ...
0
votes
0answers
23 views

When is the LU decomposition unique?

I want to find out when a matrix decomposition $A = LU $ (L lower and U upper matrix) is unique? Clearly, if $A$ is not invertible, there is no chance that this decomposition is unique. Hence, ...
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votes
1answer
32 views

Symmetric matrix problem

$A$ is a symmetric matrix and has a eigenvalue $\lambda$ of order m why $\lambda$ has m independent eigenvector
3
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1answer
40 views

Norm of symmetric positive semidefinite matrices

I have been researching a lot trying to find an answer to my question and didn't find any so I would appreciate it if anyone can help. If we have 2 symmetric, positive semi-definite matrices $A$ and ...
0
votes
0answers
25 views

Error in Matlab code, help please

I have a Unitary matrix MM1 and a theta function , I am trying to create another analytic function f by by summing up the products MM1(i, j)*thet(j) i.e say when j =1 f(1) = (MM1(1,d)*thet(0)) + ...
1
vote
4answers
28 views

Upper bound for the rank of a nilpotent matrix , if $A^2 \ne 0$

I came across the fact that the rank of a nxn-matrix A with $A^2=0$ is at most $\frac{n}{2}$. The easiest way to proof this is using the inequality $rank(A) + rank(B) -n \le rank(AB)$. With $A=B$ and ...
0
votes
0answers
18 views

Is it a Wishart matrix? [duplicate]

Now I am having some problems about the Wishart matrix. Please help me, thank you! We know that $m \times m$ random matrix $\boldsymbol{A} = \boldsymbol{H} \boldsymbol{H}^H$ is a (central) ...
2
votes
0answers
24 views

Is it a Wishart matrix or not?

Now I am having some problems about the Wishart matrix. Please help me, thank you! We know that $m \times m$ random matrix $\boldsymbol{A} = \boldsymbol{H} \boldsymbol{H}^H$ is a (central) ...
1
vote
1answer
13 views

Cross-product is a left singular vector?

Assume A is a 3x2 matrix with rank(A)=2. u1 and u2 are already left singular vectors... How would I go about proving that the cross-product of the two is also a left singular vector? Hints would be ...
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votes
1answer
33 views

Let 1r be the identity matrix… [on hold]

I'm just beginning this subject and finding it hard to get my head around some of the terms. Any help is appreciated.
0
votes
1answer
24 views

Cholesky factorization and non-positive definite matrices

When Cholesky factorization fails, is there an alternative method to obtain the $\mathbf{L}$ matrix in: $\mathbf{A}=\mathbf{L}\mathbf{L}^{*}$ I'm dealing with a matrix not guaranteed to be ...
1
vote
2answers
69 views

On the nilpotence of the matrix $AB-BA$ [on hold]

Given $n\times n$ matrices $A,B$ satisfy: $rank(AB-BA)=1$ Prove that $(AB-BA)^{2}=0$ Generalize the problem if possible. Any solution not mention Jordan canonical form would be appreciated!
0
votes
1answer
25 views

Convexity of trace for the product of two matrices

I have the following function for two matrices ${\bf A}$ and ${\bf B}$: $f({\bf A}, {\bf B}) = Tr\{({\bf Y - XAB)}^T({\bf Y - XAB)}\}$ where $Tr$ represents the trace function and matrices ${\bf X}$ ...
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votes
2answers
26 views

When does the system of equations have initly solution, no solutions, and only one solution [on hold]

I am trying to solve this math problem. So far I am bad at it. I nedd to determine for what given x does the system of linear equations has no solutions, has one solution, or infinetly many solutions. ...
0
votes
0answers
18 views

Finding the exponential relation between two 4x4 transition matrices

Im alright with matrices, but this question has dumb-struck me. Suppose I have two known and given $4\times4$ transition matrices, representing transitions in three dimensions with the fourth ...
4
votes
3answers
137 views

$A+A^2B+B=0$ implies $A^2+I$ invertible?

Let $A$ and $B$ be two square matrices over a field such that $A+A^2B+B=0$. Is it true that $A^2+I$ is always invertible ?
3
votes
3answers
48 views

Homotopy of Involutory Matrices?

I want to construct a homotopy from the matrix $$ \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} $$ ...
1
vote
1answer
35 views

Some “Product” of Positive Definite Matrices

I could remember that if $A,B$ are two positive definite matrices, then $(a_{ij}b_{ij})$ is positive definite also. But I could not see how to prove it then.
0
votes
3answers
62 views

About semipositive definite matrix

Suppose $A$ and $B$ are positive semidefinite matrices $A \ge B\ge 0$ Is the statement $A^2\geq B^2$ true or false? Why? $\geq$ means nonnegative pointwise
1
vote
1answer
31 views

Study endomorphism diagonalization

Given an endomorphism whose matrix is: $\begin{pmatrix} 1+a & -a & a \\ 2+a & -a & a-1 \\ 2 & -1 & 0 \end{pmatrix}$ How can I study if it's diagonalizable or not depending ...
1
vote
0answers
12 views

Fast way to find exponential of a matrix dot product where one of them is diagonal

Suppose $Q$ is a dot product of diagonal matrix A and matrix B: $$ Q=A\cdot B= \left( \begin{matrix} a_1 & 0 & \cdots & 0 \\ 0 & a_2 & \cdots & 0 \\ ...
0
votes
0answers
19 views

Explain why a matrix is orthogonally diagonalisable.

If people could tell me if I'm on the right track on and give me a push in the right direction for the ones I'm unsure of that would be much appreciated. Let A$\epsilon$M$_{3}$($\mathbb{R})$ and ...
2
votes
1answer
38 views

Matrix time derivative

Given a complex, square matrix $A$ that is diagonalizable, is it possible to write a simple formula for $\frac{d}{dt} A^t$ for a real, positive power $t$ and for $A$ a smooth function of $t$?
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0answers
18 views

Finding true bearings?

What is the true north bearing of NNE on 16 point cardinac compass? I just wanna know that is there any exact bearing or do we have to only give an approximate bearing?
0
votes
1answer
22 views

A and B are nxn matrices. A = $B^{T}B$ Prove that if rank(B)=n, A is pos def, and if rank(B)<b, A is pos semi-def.

A and B are nxn matrices. A = $B^{T}B$ Prove that if rank(B)=n, A is positive definite, and if rank(B) My current understanding is that if rank(B)=n, then rank($B^{T}B$)=n then rank(A)=n, making A ...
0
votes
3answers
26 views

Prove that $b^2 pr =q^2 ac$ using matrices

Let $i_1,i_2$ and $j_1,j_2$ be non-zero real roots of $ax^2+bx+c$ and $px^2+qx+r$ respectively, where a,p $\neq$0. If the system of equations $ i_1y+i_2z=0$ and $j_1y+j_2z=0$ has a non-trivial ...
1
vote
1answer
33 views

Find a 2x2 matrix with positive eigenvalues, but a negative quadratic form for some x in $R^{2}$

Find a 2x2 matrix with real and positive eigenvalues, but a negative definite quadratic form. Also, find a 2x2 matrix with real and positive eigenvalues, but an indefinite quadratic form. Isn't this ...
0
votes
0answers
11 views

Cesaro limit of stochastic matrices [on hold]

For a Markov chain, we can write the transition matrix as $$ P = \left( \begin{matrix} Q & R_1 & R_2 & \cdots & R_h \\ 0 & B_1 & \mathbf{0} & \cdots & \mathbf{0} \\ 0 ...
-1
votes
0answers
17 views

Matrix Perturbation bounds for large perturbations

Suppose I have a matrix $M \in \mathbb{R}^{m \times n} $ which has a rank $r < < m,n$ . The matrix is perturbed by an error matrix $E \in \mathbb{R}^{m \times n} $, such that $||E||_{F} \leq ...
0
votes
1answer
58 views

Connection between Eigenvectors and linear equations

I'm trying to understand the connection between Eigenvectors/Eigenvalues and linear equations: $Ax=b$ If you are given the eigenvectors and eigenvalues of $A$, can you construct the solution for the ...
0
votes
0answers
33 views

show by using leibniz formula

There are given $ r, s,n \in\mathbb N$ and $r+s=n$. It also given $A \in M_{r,K} $, $B \in M_{r\times s,K} $ and $C \in M_{s,K} $. Let $M$ be the matrix $\begin{bmatrix}A & B\\0 & ...
8
votes
0answers
196 views

How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrix $A,B,C\in M_{n}(C)$ is Hermitian matrix and is Positive definite matrices ,such $$A+B+C=I_{n}$$show that $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge ...
0
votes
1answer
10 views

invert lower triangular matrix

I am sorry if the question is simple,am trying to find the quicker method to invert a triangular matrix. Could you please provide some references where i could refer? Moreover,is there any known way ...
1
vote
1answer
34 views

Solving an augmented coefficient matrix so there are infinitely solutions

I am trying to figure out this math problem. For what values $a,b$ does the linear system have infinitely many solutions? This is the matrix $$ \left[ \begin{array}{ccc|c} ...
0
votes
1answer
22 views

what is the name of this matrix? does it have any special characteristics?

does anyone know the name of this matrix or if it has any special characteristics or how to calculate its inverse efficiently e.g. in a closed-form? [ \begin{array}{llllll} ...