Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

0
votes
1answer
23 views

Eigenvalues of an antihermitian matrix

I have to prove that every eigenvalue of an antihermitian matrix is in the form of $bi $ for some $b \in R$. I already know that if A is antihermitian , it is normal , thus we can diagonalise it ...
0
votes
0answers
7 views

standard definition and notation for a truncated permutation

I am seeking the standard notations or the standard way to define a truncated uniform random permutation. Let $M$ be an $n \times n$ matrix where each row and each column contain a single 1 and all ...
0
votes
0answers
14 views

Non-Negative Vs Positive Semi Definite

A matrix is PSD if $$\langle Ax, x\rangle \ge 0, \forall x \in H$$ Where, H is a hilbert space and A is a mapping $H \rightarrow H$. Is it the same as being Non-negative? I couldn't seem to find a ...
0
votes
3answers
57 views

Inverse Matrix in Python

My understanding is that I can use Python to initialize my matrix and then apply an inverse function to find the solution.
1
vote
0answers
9 views

Counting the powers of a companion matrix that possess nonzero leading principal minors

Let $p$ be prime, let $A$ be the $n\times n$ companion matrix associated to a primitive polynomial $f(x)$ of degree $n$ with coefficients in ${\mathbb Z}_p$, and let $T=\{A^j\mid 1\leq j\leq p^n-1\}\...
0
votes
0answers
37 views

Changing rotation center

Things that we have: 2 dimensions, a object with it's coordinates (object P1), it's rotation center (pivot) C1. After that lets rotate it at pivot C1 by known angle A. Now let's move that pivot by ...
1
vote
1answer
25 views

Generate a matrix whose entries are $0$ and $1$ such that all its submatrices are not equal

Given $m$, $n$. Are there any methods for generating an $m\times n$ matrix whose entries are $0$ and $1$ such that all its submatrices are not equal? For example, all submatrices of the following $4\...
0
votes
0answers
38 views

complexity of QR decomposition .

Could anyone help me to find the complexity of QR decomposition of matrix $A \in C^{N \times P}$, where $P \leq N$. I am also willing to know the complexity of adding a matrix $A \in C^{N \times N}$ ...
1
vote
1answer
35 views

When do matrices respect inequalities?

Suppose I have vectors $u, v \in \mathbb{R}^n$ such that $u \le v$, meaning that, component-wise, each element of $u$ is less than or equal to the corresponding component in $v$. I am curious about ...
1
vote
0answers
48 views

How to calculate determinant of $n \times n$ matrix [duplicate]

How can I calculate the determinant of the following matrix? $$A=\begin{pmatrix} \lambda&&&a_0\\ -1&\ddots&&\vdots\\ &\ddots&\lambda&a_{n-2}\\ &&-1&\...
0
votes
1answer
17 views

Correct notation for broadcasting operation

In Python numpy a row vector $u \in \mathbb{R}^n$ and its transposed $u^T$ can be added, multiplied or subtracted, yielding a $\mathbb{R}^{n\times n}$ matrix, where each element $(i,j)$ is the ...
0
votes
2answers
71 views

Solve $x = Cx+d$ for $x$

Given $$C = \begin{bmatrix} 0.7 & 0.2 \\ 0.1 & 0.5 \\ \end{bmatrix} \qquad \text{ and } \qquad d = \begin{bmatrix} 26\\ 52\\ \end{bmatrix}$$ Solve the equation $x = C x + d$ for $...
1
vote
1answer
35 views

Determinant of a matrix with cofactor expansion

I have this question here: Suppose that $A$ is an $(n,n)$-matrices, $n \geq 3$, with $\det(A)=5$ and whose $(2,3)$-minor $M_{2,3}(A)$ equals $12$. Let $S_{2,3}$ be the $(n,n)$-matrix all whose ...
0
votes
0answers
14 views

Is it possible to simplify $\text{Tr}[(\Omega A)^2]$?

The $3\times3$ matrix $\Omega$ is skew-symmetric with the determinant equals to zero. The matrix $A$ is symmetric and is non-singular. Is it possible to have the expression $\text{Tr}[(\Omega A)^2]$ ...
-1
votes
3answers
45 views

Symmetric Matrices with zero eigenvalues

Is it possible to have a symmetric non zero matrix that has all its eigenvalues equal to zero?
0
votes
0answers
14 views

Are the generalized $\lambda$ eigenvalues of a matrix T and its transpose the same?

I know that the eigenvalues of a matrix $T$ and its transpose $T^T$are the same, but is this true for the generalized eigenvalues? i.e. if a matrix $T$ has $\lambda$ as a generalized eigenvalue, is ...
0
votes
1answer
16 views

Properties of matrices satisfying certain equations

I am trying to show that if a matrice $C$ is such that for all $\sigma \in \mathfrak{S}_n$, $$ \displaystyle \prod_{i=1}^n c_{i \sigma(i)} = 1 $$ then there exists diagonal matrices $A = \text{diag}(...
0
votes
1answer
40 views

Linear combination of the rows of a matrix with all resulting components different to zero

I have a matrix with n rows and every column has at least one non-zero element. I want to produce a linear combination of the rows (preferably one that contains the smallest coefficients possible) ...
1
vote
1answer
82 views

How can I show that this matrix has no inverse?

Let $A = [a_{ij}]$ be an $n \times n$ matrix with entries in $\mathbb{R}$. Suppose there exists an $m$ with $a_{ij} = 0$ for $i \ge m$ and $j \le m$, and $a_{i,i} \ne 0$ for $1 \le i \lt m$. Show that ...
-1
votes
3answers
39 views

$Ax=0$, $x \neq 0$ implies $A$ is singular

Where $A$ and $x$ are matrices. A proof of something relies upon this statement, but it's brushed over so quickly that it must be nearly trivial, I can't see it though!
-2
votes
0answers
20 views

How would I work out how many scalar multiplications I'll need? [closed]

I have a question on scalar multiplications and I have to work out how many scalar multiplications I'll need for two subscripts. Does anyone know how i would work this out? The two subscripts are: $$...
1
vote
0answers
19 views

Characterisation of Permutation Matrices

$\newcommand\mat{\mathbf}$A permutation matrix is a matrix whose columns are a permutation of the columns of the identity matrix $\mat I$. In other words, a permutation matrix is a matrix $\mat P$ ...
4
votes
1answer
54 views

If A is a real non-singular square matrix, then there exists a real matrix $B$ such that $e^B$ $=$ $A^2$

If we consider the matrix exponential map on $M_n(\mathbb R)$, then what will be the image set of the exponential map? I have seen this. From there I can say that $exp(M_n(\mathbb C))=GL_n(\mathbb C)$...
0
votes
1answer
27 views

SVD and PCA - help

I have some data on cars: i have been given the Noise,size,speed,if its electric or not, if its a lorry or not. it looks something like this: $$ \begin{matrix} Noise & Size & Speed &...
4
votes
3answers
60 views

Eigenvalues of a $A^T A$

Given the matrix of order $1\times{n}$, $A=(a_1, a_2, ..., a_n)$ , where $a_i$ are real; The question is to find all eigenvalues of $A^T A$. I have proved that it is a non-invertible matrix, ...
1
vote
0answers
29 views

Are the invariants related in the characteristic equation of an orthogonal matrix (3x3 matrix)?

The characteristic equation of matrix A is $$\lambda ^3 - I_1\lambda^2 + I_2\lambda-I_3 = 0 $$ For orthogonal matrix $$I_3 = det(A) = \pm1$$ $$I_1 = tr(A)$$ Taking examples of orthogonal matrices, ...
0
votes
0answers
23 views

Solve for matrix in squared formula

Assuming $b$ is a scalar, $v$ a vector and $S$ is a symmetric matrix. How do can I express $S$ in terms of $b$ and $v$ if $$ b = v^T\cdot S^{-1}\cdot v $$ Note that $b$ and $v$ are given and non-...
2
votes
1answer
22 views

Implicit multivariable differentiation with Jacobian matrix

Two functions $u = u(x,y)$ and $v = v(x,y)$ are defined by the system of equations: $$u-v=x-y$$ $$yu-xv=1$$ The problem asks for the partial derivatives $\frac {\partial u} {\partial x},~ \frac {\...
2
votes
0answers
27 views
+200

Norm minimization with Lie theory

Consider the following flow on square matrices. $$Y(t)= \frac{d}{dt}X(t)$$ I would like to find a skew-Hermitian matrix $K$ which minimizes the following under the Frobenius norm. $$||[K,X]-Y||_F$$ We ...
1
vote
0answers
20 views

Relationship between eigenvalues(or spectrum) of graph Laplacian matrix and the eigenvalues of product of a matrix and the Laplacian matrix.

Is there any relationship between eigenvalues(or spectrum) of graph Laplacian matrix and the eigenvalues of the product of a matrix and the Laplacian matrix? For example, let the Laplacian matrix be $...
0
votes
0answers
14 views

Pseudo determinant of a non negative matrix (singular matrix here ).

This wikipedia article gives the following formula for finding the pseudo determinant of a matrix. $$ |A|_+ = \lim\limits_{\alpha\to0} \frac{|A+\alpha I|}{\alpha^{n-\mathrm{rank}(A)}} $$ where $A$ is ...
2
votes
2answers
50 views

How to remove the parentheses of the matrix product $( (AB)^T (CDE)^T )^T$

How to remove the parentheses of the matrix product $( (AB)^T (CDE)^T )^T$ I recently started doing a course relating to IT practices and as an extra challenge we were told to solve this equation. It ...
1
vote
1answer
36 views

Gradient of Quadratic Form with Inverse of Complex Matrices

I want to calculate the gradient of $$ w^H H F (F^H F)^{-1} F^H H^H w $$ with respect to $ F $, which is complex. I am basing on this previous answer Derivative of Nested Matrix Quadratic Form ...
5
votes
1answer
56 views

An inequation about real pairwise commuting matrices

Let A and B be two real $n\times n$ matrices such that $AB=BA$. It’s known that $\det(A^2+B^2)\geq 0$. I wonder if it’s true that: For $k$ pairwise commuting real matrices $A_1,\cdots,A_k$,we have: $...
0
votes
1answer
32 views

Determinant of a particular large matrix

I've been trying to solve this problem but I stucked. Let $A = [a_{ij}]$ with size $ 2011 \times 2011$ , and given the condition below \begin{equation}a_{ij}= \begin{cases} (-1)^{|i-j|}, &...
1
vote
1answer
39 views

How to perform Gaussian elimination to invert a matrix if the matrix contains zeros on the diagonal?

I'm coding the method that inverts a matrix through Gaussian elimination. I've coded everything assuming that there are no zeros on the diagonal. For the situation where a diagonal element is zero, ...
0
votes
1answer
37 views

Prove that $A^{-1} + B^{-1}$ nonsingular by showing that $(A^{-1} + B^{-1} )^{-1} = A( A + B )^{-1} B$

Let $A$, $B$, and $A + B$ be nonsingular matrices. Prove that $A^{-1} + B^{-1}$ is nonsingular by showing that $( A^{-1} + B^{-1} )^{-1} = A( A + B )^{-1} B$ I have done progress to only knowing ...
0
votes
0answers
16 views

Centering a polygon on the origin by affine translation

Given a set of tuples, each representing the vertices of a polygon, I would like to center it on the origin. Having calculated the centermost point of the polygon (not the arithmetic centroid) and the ...
0
votes
0answers
19 views

Triangle Inequality for p-norm on nxn real matrices

Consider the following norm: $$ ||A||_F = \left( \sum^m_{i=1} \sum^n_{j=1} |A_{ij}|^p \right)^{1/p}. $$ I would like to prove that it is indeed a norm by proving the triangle inequality: $$||A+...
1
vote
1answer
23 views

System of equations can be interpreted as intersection of $3$ planes in $3$-dimensional space

I'm struggling with this question. I have worked out that attempting to solve for the $x_i$ leads to a contradiction, and that $$\begin{vmatrix}1&4&6\\1&-2&1\\2&14&17\end{...
-4
votes
1answer
32 views

Linear Algebra Ordered Basis

If $V$ is a subspace of $\Bbb{R}^3$ with ordered basis $B = (v_1,v_2)$, and let $v_3,v_4 \in V$. $A$ is the $3 \times 4$ matrix whose $i$th column is $v_i$. Find a $3 \times 2$ matrix $B$ and a $2 \...
0
votes
1answer
20 views

Closed form for matrix power sum involving product and transpose

Given square, conformable matrices $\mathbf{M}$, and $\mathbf{C}$, I'm looking for a closed form for $$\sum_{i=0}^{n} \mathbf{M}^{i} \mathbf{C} \left(\mathbf{M}^{i}\right)^{T}.$$ Or, alternatively,...
1
vote
1answer
20 views

Intuition behind finding projection matrices of transformation

Supposedly a matrix $A$ is diagonalisable iff it can be written $A=\sum \lambda_i \pi_i$ where $\lambda_i$ are the distinct eigenvalues of $A$, and $\pi_i$ are projections satisfying $i \neq j \...
1
vote
3answers
65 views

What does the notation $A^{-2}$ mean if $A$ is a matrix?

$A^2$ means to multiple the matrix by itself, and $A^{-1}$ refers to the matrix's inverse. Would $A^{-2}$ be the square of the inverse or the inverse of the square?
0
votes
1answer
31 views

For which values of L does the vector [l,3,5] belong to span {[1,0,-2],[-3,1,7]?

I'm not sure how to solve this problem. So far I have created the system of equations $c_1 -3c_2 = l$; $c_2 = 3$; $-2c_1 + 7c_2 = -5$. However, I cannot figure out how to solve the set of ...
0
votes
0answers
9 views

Matrix representation of a finite difference with Neumann boundary conditions

Given 1D data $[c_1,c_2,c_3,\cdots,c_N]$ I can represent the derivative operation as a matrix product. For example, using the central difference $$ \left.\frac{\partial c}{\partial x}\right|_k \...
0
votes
0answers
42 views

Prove that if matrix $C = I - 2M $ and $M = M^2, $ then $ C^3 = C.$

I'm faced with a problem for which I haven't been given a correction, so I expected you could tell me if I am right or not, and in the later case give me the appropriate answer. Let $M$ be an ...
-2
votes
0answers
16 views

Is it possible to convert world coordinate to screen coordinate without using matrices?

Out of curiosity, I'm wondering if it is possible to calculate screen coordinate from world coordinate without using matrices. To calculate the screen coordinate, I need Model-view(Camera, scale, ...
2
votes
1answer
34 views

Question on the definition of an Inverse matrix [duplicate]

By definition, if $A$ is a $ n \times n $ matrix, an inverse of $A$ is an $ n \times n $ matrix $A^{-1}$ with the property that: $$ A^{-1}A=\mathbb I_n \ \ \land \ \ AA^{-1}=\mathbb I_n \ \ \ \ (1)$$ ...
0
votes
0answers
6 views

Estimation of eigenvalues for online convergence estimation

Given some matrix $A$ of the form: $$ A \equiv \left(H^\mathrm{H}\Sigma H + \Lambda\right) $$ with $\Sigma$ and $\Lambda$ full-rank diagonal real matrices, and $H$ a large rank deficient not-...