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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.

2
votes
3answers
21 views

X is to vector like matrix is to linear operator?

In linear algebra texts there is usually a clear distinction between linear operators and matrices. A linear operator is a map between two spaces that fulfills a set of conditions. A matrix is a 2D ...
0
votes
4answers
32 views

If $M \ne N^2$ is $\det(M-N^2)\ne 0$?

Does $M \ne N^2$ imply that $\det(M-N^2)\ne 0$? I believe it's correct because: $M- N^2 \ne O$ where O is null matrix and after taking determinant on both sides we get $\det(M-N^2) \ne 0 $ but I am ...
2
votes
0answers
23 views

How do I solve $\mathbf{x}^T e^{t\mathbf{A}}\mathbf{y}=c$ for $t$?

I have the following matrix equation: $\mathbf{x}^T e^{t\mathbf{A}}\mathbf{y}=c$ $\mathbf{x}$ and $\mathbf{y}$ are vectors of length $k$, $\mathbf{A}$ is a $k\times k$ matrix, and $c$ and $t$ are ...
0
votes
1answer
24 views

Can i multiply an [6x1] matrix with an [6x6] in linear algebra?

I wish to multiply the first matrix float X[6] = { x, //Position y, //Position z, //Position _x, //Velocity _y, //Velocity _z //Velocity }; with ...
-1
votes
0answers
31 views

The matrix exponential

Prove that $$(e^A)^{∗}=e^{A^∗} .$$ I've tried messing around with both sides, I just can't get the two to match up. Any ideas?
0
votes
1answer
30 views

Calculation of Matrix

I would like to know how to calculate this. For example, in row, it shows size (SS,S,M and L) and in column, it shows colour (red, blue, yellow). If i calculate the number of units in each size and ...
0
votes
2answers
38 views

Show that a linear map on a finite dimensional complex vector space always have an eigenvalue.

What is an alternative proof that a linear map $T$ on a finite dimensional complex vector space $V$ with dimension $n$ always has an eigenvalue? Here is the original proof idea: We take a no zero ...
2
votes
0answers
18 views

Criterion that unique solution of system of linear equations with integer coefficients is integer (relation to Vandermonde matrices)

Let $2\leq p\in\mathbb{N}$, $r\in\{0,\ldots,p-1\}$, $k,d\in\mathbb{N}$ with $k\leq d$, \begin{align} A&:=\left((r+ip)^j\right)_{i,j\in\{0,\ldots,k-1\}}= \left(\begin{matrix} (r+(\rlap{0}\phantom{k-...
0
votes
2answers
26 views

Eigenvalues and eigenvectors for the moment of inertia matrix

Find the eigenvalues and eigenvectors for the moment of inertia matrix given by $$I={m\over 2}\left(\begin{matrix} 1 & -1 & 0\\ -1 & 1 & 0\\ 0 & 0 & 2\end{matrix}\right)$$ ...
0
votes
0answers
18 views

Derivative of scalar wrt vector and derivative of scalar wrt matrix

It seems to me that the derivative (or more precisely) the gradient of a scalar valued function with respect to (wrt) a vector can, at least in most contexts be considered equivalent to the gradient ...
0
votes
3answers
36 views

Invertibilty of the inverse matrix

Suppose I have a matrix $A$ which is invertible. Call $A^{-1}$ the inverse of $A$. Is $A^{-1}$ invertible just because is the inverse of another matrix? Suppose also I know the determinant of $A$, $...
1
vote
1answer
17 views

Apply the Hadamard transform to different state vectors

I am in the process of understanding a proof. First, the following is said there: $$H\begin{pmatrix}1\\0\\\vdots\\0\end{pmatrix}=\frac{1}{\sqrt{N}}\begin{pmatrix}1\\1\\1\\1\end{pmatrix}$$ This is ...
0
votes
1answer
16 views

How to create a transformation matrix for a M22 → M22 transformation

I have a linear transformation, T, such that; T:${M_{22}}$→${M_{22}}$: T$\left(\begin{bmatrix}{x_{11}} & {x_{12}}\\{x_{21}} & {x_{22}}\end{bmatrix} \right)= \begin{bmatrix}{{x_{12}}-5{x_{21}...
1
vote
0answers
56 views

Product of matrices and system of equation

I have tried everything in my power but I still cant figure how to do this :/ $$AB=\begin{bmatrix}1&2\\3&4\end{bmatrix}$$ How can I show that if $A$ is a $2\times3$ matrix, then the system: $...
2
votes
0answers
9 views

Harmonic Analysis on a Finitely-Generated Matrix Group

Let $\mathbf{M}_{1},\ldots,\mathbf{M}_{N}$ be invertible $2\times2$ matrices with entries in $\mathbb{Q}$ of the form: $$\mathbf{M}_{n}=\left[\begin{array}{cc} a_{n} & b_{n}\\ 0 & c_{n} \end{...
1
vote
0answers
31 views

a differential equation with matrices

How can I solve this system of equation: $$ Mx''(t)+(K+N)x'(t)+(D+G)x(t)=F(t) $$ $ x(0)=0 $ where M, K, N, D, G are matrix I tried substitution $ z=(x, x') $
0
votes
0answers
14 views

Which Covariance Matrix to Use for Mahalanobis Distance Between Two Groups?

I have two groups of data with the same number of variables. One group has a disease and one group does not. I am taking the pair-wise mahalanobis distance of each person in the disease group from the ...
3
votes
3answers
34 views

Find the values of a and b that make the equation system have infinite solutions

I'm given the following system of equations and I need to find the values for a and b so that the system has infinite solutions. \begin{cases} &3y &- &2z &= 5 \\ &-2x &+ &...
0
votes
1answer
18 views

Lowest Possible Average Correlation Between N Random Variables

Suppose $M$ is an $n \times n$ correlation matrix, with correlation $\rho_{i,j}$ between any pair of two random variables. What is the smallest possible average of the $\rho_{i,j}$ where $(i<j)$? ...
0
votes
1answer
22 views

Kronecker product on a matrix with structured blocks

I'm currently attempting to write a symmetric matrix with structured blocks into Kronecker-factorized form, but I'm not sure if the task is possible at all. My matrix takes the following form: $$ M= \...
0
votes
1answer
25 views

Proof the question of marix [duplicate]

Prove that that a system of linear non homogeneous equation AX=B is consistent if rank of the matrix A equal the rank of the augmented matrix(AB).
0
votes
0answers
26 views

Solution to first order ODE system

I have an equation which describes how wave modes evolve spatially, according to \begin{equation} \frac{\partial \mathbf{P_{\omega}} (0,x_3)}{\partial x_3} = \begin{bmatrix} \Delta_1(x_3) & \...
-2
votes
2answers
33 views

I wants answer of this question [on hold]

Prove that the rank of a matrix does not change after pre multiplication or post multiplication by any non singular matrix
1
vote
1answer
24 views

Moore-Penrose pseudoinverse and multiplication by diagonal matrix

Let $A \in \mathbb{R}^{n \times p}$, let $D$ be a diagonal matrix with positive entries. $\dagger$ denotes the Moore-Penrose pseudoinverse. Is it true in general that: $$(A^\top D A)^\dagger A^\top D ...
1
vote
1answer
42 views

Characteristic polynomial and eigenvector of Frobenius matrix

Consider the following $n \times n$ matrix (I believe this is similar to companion matrix): $$ A = \begin{pmatrix} 0 & -1 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 0 & -...
0
votes
0answers
37 views

prove the rank of $AB$ [on hold]

1. $A$ is $m$ by $n$ matrix and $B$ is $n$ by $p$ matrix. Prove that $AB$ can be written as a sum of n matrices of rank at most one. $A$ is $m$ by $n$ matrix with rank $m$ and $B$ is $n$ by $p$ ...
0
votes
0answers
14 views

Prove properties of the conditional inverse [on hold]

Let A be a $n$ x $p$ matrix with $n\geq\ p$ Show that: $r(A^CA)$ = $r(A)$ and $r(I - A(A^TA)^CA^T)$ = $n-r(A)$ thankyou
0
votes
0answers
17 views

Matrix Reduction with Positions

For a matrix $M$ and a set of positions $P=\{(x_i,y_i) \ | \ 1 \leq i \leq n\}$, define $f(M,P)$ as the sum of elements of $M$ at $P$, i.e., $f(M,P)=\sum_{(x_i,y_i)\in P}M_{x_i,y_i}$. Now given $M$ ...
0
votes
0answers
19 views

Matrix perturbation and eigenvector

$Ae=\lambda e$ $e^Te=1$ $A$ is a real matrix. $\lambda, e$ are real. $(A+\Delta A)(e+\Delta e)=(\lambda+\Delta \lambda)(e+\Delta e)$ Neglecting small terms, $\Delta Ae +A \Delta e=\Delta \lambda ...
0
votes
1answer
43 views

How many unique possibilities for $n\times n$ matrix are there?

When given a $n \times n$ matrix (for simplicity let's say $n=3$) how many unique possibilities are there to fill the entries with a set of $m$ numbers (again for simplicity let's say $m=10$). With ...
1
vote
1answer
34 views

Does the following result hold? [duplicate]

Suppose that $A$ and $B$ are symmetric and non-negative matrices. Let $\rho(A)$ denote the spectral radius of $A$ and $\rho(B)$ denote the spectral radius of $B$. Does the following result hold? $...
0
votes
2answers
37 views

Given path of a unit vector can we derive the rotation matrix?

Let's say a unit vector $\overrightarrow{A}(t)$ in N dimensions is continuously rotated around the origin. It may for example trace out a circle in 3 dimensions or some sort of spiral in higher ...
0
votes
0answers
11 views

Prove that $\langle\mathbf{A}, [\mathbf{C}; \mathbf{0}]\rangle \leq \delta$ equals with $\|\mathbf{A_r}\|_*\leq\delta$ [on hold]

Given an arbitrary matrix $\mathbf{A}\in R^{n\times n}$ and the basis matrix set $\mathbb{S}=\{\mathbf{C}\in R^{r\times n}: \mathbf{C}\mathbf{C}^T=\mathbf{I}\}$. $[\mathbf{C}; \mathbf{0}]\in R^{n\...
1
vote
1answer
15 views

Finding matrix from plane in kernal

Give an example of a matrix A such that $\ker(A)$ is the plane $2x − y + 3z = 0$. I am not sure where to start, as I know that the $\ker(A)$ is the matrix of the plane, but I don't know how to go ...
0
votes
2answers
29 views

Given two unit vectors what can we say about the rotations that can transform one into another?

Given 2 unit vectors in 2 dimensions there is a uniqe rotation matrix that transforms on to the other. $A$ and $B$ are the two unit vectors is there a general way to find the set of orthogonal ...
1
vote
1answer
27 views

How to compute the derivative $f(X) = \|\mathcal{P}_\Omega(X-A)\|^2_F$?

How to compute the derivative $$f(X) = \| \mathcal{P}_\Omega(X-A)\|_F^2$$ here $\mathcal{P}_\Omega(\cdot)$ is a projector, $[\mathcal{P}_\Omega(Y)]_{ij} = Y_{ij}$ if $(i,j)\in \Omega$, zero otherwise....
0
votes
1answer
45 views

Why are singular values of “complex” matrices always real and non-negative?

I've already read the following related questions on math.SE: Why can't singular values be complex numbers? Clarification on the SVD of a complex matrix Why are singular values always non-negative? ...
0
votes
0answers
11 views

What can be said about the definiteness of the following inequality?

Given a Hurwitz matrix $R\in\Re^n$ which has all the eigenvalues located in the closed left-half plane. For a positive-definite matrix $Q$, we know that there exists a unique solution $P$ to the ...
2
votes
1answer
36 views

Normalizer in matrix groups

I have the problem of calculating the normalizer of $\begin{bmatrix}      \lambda & 0 \\ 0 & \lambda^{- 1} \end{bmatrix} $ in the group $\begin{bmatrix}      \cos (\theta) & -\sin (\theta) ...
0
votes
1answer
37 views

States of the world/Game theory and Beliefs

This post consists on 3 parts: the question itself, hint and a table. The question will make sense to you only after you have read the tables and the hint attached. The problem is about beliefs of a ...
0
votes
2answers
22 views

What is meant by tridiagonal linear equation system?

I have to implement the SOR (Successive Over-Relaxation) method, using sparse matrices, to find the solution vector of these linear equations systems (for quite huge matrices): What does that tridiag(...
0
votes
0answers
22 views

Matrices up to an equivalence relation

Consider the $n\times n$ matrices over a field $k$, $M_{n\times m}k$. Say that $A\sim B$ for $A,B\in M_{n\times n}k$ if there is an invertible matrix $C\in M_{m\times m}k$ such that $A=BC$. Under ...
1
vote
0answers
33 views

Matrix of integer powers

Is there a name for the square matrix ($j=0...n$, $k=0...n$) $M_{jk} = j^k$ (with special case $M_{00}:=1$) and is there a closed general formula for its inverse? I have stumbled upon this while ...
1
vote
1answer
13 views

Interpretation of vectors in dual forms - in matrix equation, and in linear combination of vectors

While a matrix equation $A \vec x=\vec b$ identifies $\vec x$ and $\vec b$ as two vectors, its equivalent form as linear combinations of vectors ${x_1} \vec {a_1} + {x_2} \vec {a_2} = \vec b$ reveals ...
0
votes
2answers
90 views

How to prove that a $3\times 3$ matrix has only $2$ eigenvectors?

I am working through a problem in Riley, Hobson and Bence (Mathematical Methods for Physics and Engineering) that revolves around the following matrix: $$ A= \begin{pmatrix} 2 & 0 & 0 ...
0
votes
0answers
13 views

Perform Singular Value Decomposition of the following matrix?

Perform Singular Value Decomposition of the following[2*2] matrix and represent in UDV(power)T? [3,2,2; 2,3,-2]
3
votes
2answers
58 views

Prove that the equation $ax = b$, $xa = b$, always has a unique solution. [on hold]

A vector space $A$ is called an algebra if in it, in addition to the addition of vectors and multiplication by a number, the multiplication of vectors with properties is defined. In other words, for $...
0
votes
0answers
23 views

Least squares solution to overdetermined AX=B where each matrix is a rotation

I can't seem to find anything that would help me with this particular problem. I have a bunch of measurements of a matrix A and corresponding matrix B, which I know are related by a third rotation X (...
0
votes
1answer
12 views

Most elegant way to show Inner-Product of tensors with indices?

How would you write this sum most elegantly? (Or in other words: What is the most elegant and best way to show a sum of products with indicies?) $\sum_{k} A_{a_{1},...,a_{i},k}*B_{k,b_{1},...,b_{j}}$ ...
0
votes
1answer
43 views

I would like to know how to calculate Matrix

$\begin{pmatrix} x & y+2 \\ 5z & 9k \end{pmatrix}=\begin{pmatrix} 2 & 8 \\ 3 & 9 \end{pmatrix}$ This is calculation of Matrix. $x= \frac{1}{3}, y=y+\frac{1}{4}, z= \frac{5}{3z}, k=...