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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0answers
12 views

Comparision between Hamming distance and cosine similarities?

I want to check the similarities between binary vectors of different length and I am using cosine similarities and hamming distances for calculations. These are of length 1000 elements(0 and 1). ...
1
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2answers
17 views

matrix-finding determinant of adj of inverse matirx

if A is a $3$x$3$ matrix and let A=$2$,then what will be the value of det(adj(adj(adj($A^{-1}$)))? 1.$\dfrac{1}{512}$ 2.$\dfrac{1}{1024}$ 3.$\dfrac{1}{128}$ 4.$\dfrac{1}{256}$
0
votes
0answers
25 views

Derivative with respect to a function

We have a function ${f(s,{\psi(s)}_{3\times 1})}_{3\times1}\tag1$ Given Data $f,\psi$ are matrices and their dimensions are already given in the question s is not a matrix, it is a scalar ...
0
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0answers
13 views

Mapping from $\left(-\infty,\infty\right)$ to $[a,b]$ to reduce numerical error

Suppose $A = \left[\begin{array}{cc} \exp\left(x_1\right)&\exp\left(x_2\right)\\ \exp\left(x_3\right)&\exp\left(x_4\right) \end{array} \right]$, where each of $x_i\in\left(-1000,1000\right),$ ...
0
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0answers
39 views

Function with constant derivative

We have a column matrix $P_i$ defined as follows $P_i= {\begin{pmatrix} a_i \\ b_i \\ c_i \end{pmatrix}}_{3\times 1}\tag 1 $. Given Data All $a_i,b_i,c_i$ are constants It is given that $i$ can ...
0
votes
2answers
31 views

Same column space is equivalent to same row space?

If $A$ and $B$ are $n \times n$ matrices that have the same column space, then $A$ and $B$ have the same row space. Can one prove or disprove this? This is my continuation of Same row space is ...
0
votes
1answer
39 views

Same row space is equivalent to same column space?

If $A$ and $B$ are $n \times n$ matrices that have the same row space, then $A$ and $B$ have the same column space. This is false of course. I could just come up with examples though. Can one prove ...
-1
votes
3answers
65 views

Find an invertible non-diagonal $3 \times 3$ matrix $D$ such that $D^3 = D$.

Find an invertible non-diagonal $3 \times 3$ matrix $D$ such that $D^3 = D$. I have forgotten how to solve this kind of question, can somebody give me some hint or idea how to start?
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0answers
17 views

How to perform parallel cholesky factorisation

I am a student trying to understand and implement a parallel cholesky factorisation algorithm for sparse matrices in software. Can anyone give an example of a parallel cholesky factorisation ...
2
votes
3answers
50 views

A linear map that is multiplication by a matrix

The problem statement, all given variables and data Let $T$ be multiplication by matrix $A$: $$A= \begin{bmatrix} 1 & -1 & 3 \\ 5 & 6 & -4 \\ 7 & ...
0
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3answers
56 views

Proof concerning matrix composition.

I have a statement which I don't know how to prove. All matrices are real, $n \times n$. For all $0 < k < n$ the following has to hold. It is impossible do define a matrix $A$ of rank ...
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0answers
17 views

Bounding the norm of the product of random PSD matrices

Consider the following setup, $(X, \hat{X}, Y, \hat{Y})$ are four $n \times n$ real, symmetric, full-rank, positive-definite matrices with entries between zero and one and operator norm $O(n)$. The ...
0
votes
1answer
19 views

Expectation of an exponentiated quadratic form

Given a multivariate normal random $n\times 1$ vector $X \sim N(\mu,\Sigma)$, what is the expectation $$\mathbb{E}[exp(X^TAX+b^TX)]$$ where $A$ is a $n\times n$ matrix and $b$ is a n-dimensional ...
1
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0answers
19 views

A characterization of a certain family of matrices in terms of another matrix.

Consider a real matrix $A$ of dimension $n \times n$. Assume $k \leq n$ is given. I am looking for ways to describe the following set of matrices in terms of properties of $A$. $\mathcal{S}(A) = \{B ...
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vote
2answers
77 views
+50

Lower and upper bound for the largest eigenvalue

We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius ...
2
votes
1answer
35 views

Perron–Frobenius theorem

What is exactly the Perron–Frobenius theorem? In different books papers I read different statments, and I don't know what is the truth. In wikipedia there are also a lot of statements under this ...
8
votes
1answer
106 views
+50

Theorem about positive matrices

We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius ...
1
vote
1answer
67 views

Is it possible to back solve this matrix?

After applying Gaussian reduction mod 2, I've ended up with this matrix: $$\left(\begin{array}{ccccccccc} 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & ...
0
votes
0answers
19 views

How do you find the change of coordinates matrix from B to C (defined below)?

B = {b1, b2} C = {c1, c2} b1 = 5c1 - 3c2 b2 = 8c1 - 5c2 How do you find the change of coordinates matrix from B to C? Also, how would you find [x]c for x = -4b1 + 3b2? I assumed the change of ...
-2
votes
1answer
35 views

matrix multiplication, reverse order; should comformability criteria be changed?

It concerns with 'reverse' order of matrix multiplication as stated in the book 'Computational and Algorithmic Linear Algebra and n-Dimensional Geometry' by "Katta G. Murty", in Section 2.5, titled ...
0
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0answers
11 views

Constructing an oblique projection via formula.

Assume $\Phi$ is an arbitrary given $n \times k$ real matrix with $k < n$ and with independent columns. Consider the family of oblique projections on the column space of $\Phi$. All members of ...
1
vote
1answer
53 views

Why must $b=0$ for this linear system to have infinitely many solutions for all $a$?

Consider the parameterized linear system of equations represented by the augmented matrix: $$ \left[ \begin{array}{ccc|c} 1 & 0 & a & 1 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & ...
-2
votes
1answer
30 views

Proving result on matrix rank

Is it true that, if $A=QR$ with $Q$ unitary matrix and $R$ an upper triangular matrix, and $A\in\mathbb{C}^{n\times n}$, then the rank of $A$ is the same as that of $R$? And if so, how do I prove it?
3
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2answers
91 views

Minimum linear subspaces cover problem

Given a set of vectors $V=\{v_1,v_2,...,v_n\}$ and $m$ vector sets $V_1,V_2,...,V_m$ ($V_i$ may not be a subset of $V$), I want to find minimum number of sets from $\{V_1,V_2,...,V_m\}$, denoted as ...
0
votes
1answer
68 views

Self-adjoint on dot product

Let be $V = M_3(\mathbb{R})$ the vector space of the real antisymmetric matrix and let be $\phi$ the scalar product defined by $\phi(X,Y) = tr(^tXY)~ \forall X, Y \in V$. Let be $A$ a symmetric ...
0
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0answers
15 views

Applications of low-rank matrix approximation

There was a similar question here Use of low rank approximation of a matrix that has unfortunately remained unanswered. Although being along the same lines, my question will be formulated in a little ...
1
vote
2answers
28 views

How do you find the change of coordinates matrix from a given matrix to the standard basis?

I'm not sure how to approach this problem. The examples I've come across on the internet show how to find the change of coordinates matrix from a matrix to another matrix, such as B to C (for ...
0
votes
2answers
37 views

What is wrong with my argument? (commutativity of matrix multiplication)

I was working on my pset, trying to prove $B^{-1}A^{-1}=(AB)^{-1}$. I proved it, but in the middle I saw something that led to this. $$ \begin{align} (AB)^{-1}(AB)=I \qquad &\text{(by the ...
0
votes
0answers
33 views

Solving system of equations in rationals

Do there exist solutions to solve system of $n-2$ equations with $n-2$ variables where $n$ is fixed even integer and $a_i,b,c\in\mathbb{Q},i\in\{0,1,2,\cdots,n-5\}$ $$\left\{ ...
0
votes
1answer
26 views

Transpose of higher dimension matrices

We all know transpose of 2D matrix A Old $A_{ij}$ will be replaced by $A_{ji}$ in the transpose matrix and vice versa Question If A is 3D matrices of $3\times 3 \times 8$ then what is old ...
0
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0answers
14 views

Question about solving convex quadratic problem with interior method

I want to find $x \in \mathbb{R}^n$ that minimizes $$f(x) = x^T Q x + g^T x$$ $$Ax \leq 0$$ where $Q \in \mathbb{R}^{n \times n}$, $A \in \mathbb{R}^{m \times n}$, $g \in \mathbb{R}^n$ and $Q$ is ...
2
votes
1answer
22 views

Solving a linear equation for a symmetric,positive matrix

Given the Problem $A x = b$ for some regular matrix $A \in \mathbb{R}^{n \times n}$ and $b\in\mathbb{R}^n$. One can compute $x$ with the Cholesky factorization in $O(n^3)$. If $A$ is known to be a ...
2
votes
0answers
17 views

Counting the operations of a problem

I have a square matrix $A\in\mathbb{R}^{n\times n}$, it has a LU decomposition. $L$ and $U$ are triangular and $L$ has ones on the main diagonal. I'm counting the number of operations for ...
1
vote
1answer
65 views

Continuity of the spectral radius

Let $M \in \mathbb{R}^{n\times n}$ be a nonnegative irreducible matrix with strictly positive entries on its main diagonal. Then $M$ is primitive and by the Perron-Frobenius Theorem we know that the ...
0
votes
1answer
47 views

Tensor product of Frobenius algebras

In proving the fact that the tensor product of any two finite-dimensional Frobenius algebras $R$ and $S$ over the same field $k$, it is usually defined a $k$-bilinear pairing $E: W×W→k$ where ...
5
votes
0answers
40 views

What is known about the eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular: How are the elements of the ...
3
votes
1answer
28 views

Most optimal way of grouping sets of game characters

I have been trying to solve this for two days now and have not come up with a good solution. Say if I have 8 character groups, like the following, how could I get them in teams of three so that all ...
0
votes
1answer
22 views

How do you find the vector x determined by the given coordinate vector and given basis B?

I saw a couple different ways to approach this problem from tutorials on YouTube, and each led to a different answer. This is what I got: 3 -4 | 5 -5 6 | 3 3 * 5 + -4 * 3 ...
1
vote
0answers
63 views
+50

Same eigenvalue spectrum with different matrices

There are two matrices $D$ and $T$ which provides eigenvalue spectrum (dispersion relation) according to $$ E(K) = \operatorname{eig}(D + T \exp(iK) + T' \exp(-iK)) $$ $$ K = 0:dK:\pi $$ Where K is a ...
1
vote
1answer
46 views

Under what conditions are the eigenvalues of a matrix finite?

Suppose we have a square matrix $A$. Under what conditions on $A$ ensure that all eigenvalues of $A$ are finite?
4
votes
2answers
53 views

Matrix with entries from $1$ to $16$, each occuring once, and determinant $40800$

In OEIS, it is claimed, that the largest possible determinant of a $4\ x \ 4$-matrix with the entries from $1$ to $16$, each occuring once, is $40800$. Unfortunately, the article does not mention a ...
0
votes
1answer
24 views

Describe the solution set of the system

Consider the linear system below: $$\begin{array}{ccccccc} x_1&-&2x_2&+&&&x_4&=&1\\ 2x_1& -& 5x_2& -& 2x_3& +& k^2x_4 &= &-2\\ ...
4
votes
2answers
196 views

Prove that $A \circ B = AB$ if and only if both $A$ and $B$ are diagonal

Definition. Hadamard product. Let $A,B \in \mathbb{C}^{m \times n}$. The Hadamard product of $A$ and $B$ is defined by $[A \circ B]_{ij} = [A]_{ij}[B]_{ij}$ for all $i = 1, \dots, m$, $j = 1, \dots, ...
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0answers
23 views

What to “call” an orientation matrix? [closed]

So I am programming a physics engine, and I want a good variable name for my polygons orientation matrices. often you call a matrix "u" but I want something longer, but not "orient"?
0
votes
1answer
31 views

Matrix Multiplication - When do you only multiply by one number and add vs. multiplying all numbers?

*I wasn't sure where to put this. Just let me know if I should delete it or if there is another category/website where this question would fit better. Thanks! Or if you know the answer & don't ...
1
vote
1answer
38 views

Getting matrix in row echelon form - what's my error?

I start with $\begin{bmatrix} 1 & 2k & 1 & 0 \\ 0 & 1-6k & k-3 & 2 \\ 0 & 7k & 4 & 2 \end{bmatrix}$ I want to get it in row echelon form, so I'm looking for a ...
3
votes
2answers
104 views

Inverting the infinite matrix $+\mathbf{I}$ with entries $\mathbf{P}_{ij}={i-1\choose j-1}$ [on hold]

Let $ \mathbf{P}$ denote the "infinite matrix" $$ \left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & \dots \\ 1 & 1 & 0 & 0 & \dots \\ 1 & 2 & 1 & 0 & \dots ...
0
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0answers
15 views

portfolio optimisation

I'm currently implementing a CAPM model in Excel based on the following criteria/features: A portfolio of n risky assets when n=6 (in this case) A riskless borrowing rate of 8% and riskless lending ...
2
votes
2answers
30 views

Quadratic equation not equal to zero (solving a matrix with a parameter)

I came across this in my matrix module, learning about number of solutions when the matrix has parameters. $$ \left[ \begin{array}{ccc|c} 1 & ...
1
vote
0answers
11 views

Distribution of a quadratic form

Let $A$ be a symmetric positive definite matrix, and $x$ a random vector. Suppose we know the distribution of $x^\top A x$. What can we say about the distribution of $x^\top x$?