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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.

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1answer
9 views

Given an SPD matrix, any diagonal submatrix of full rank must be SPD.

I need help with the following proof: Given a symmetric positive-definite matrix, show that any diagonal submatrix of full rank must also be symmetric positive-definite. Thanks
0
votes
1answer
25 views

Trace with respect to a variable

Supposing a matrix exponential which is a function of two sets of variables, $e^{F(\{x\},\{y\})}$ What would the meaning of $Tr_x(e^{F(\{x\},\{y\})})$ be? Terribly sorry if this is a dumb question, ...
0
votes
0answers
14 views

property of a homogeneous linear system of equations that has a trivial solution.

I am looking at a system of linear equations Ax=0 where A is a 2x2 matrix. A={{1,2},{3,4}}. I want to look at its properties. This homogeneous system has one trivial solution x={0,0}. Can I call ...
0
votes
0answers
16 views

If $M=v_1v_i^TD_i$, compute $v_i$ and $D_i$

A known real-valued $n\times n$ matrix $M$ can be written as $$ M = v_1v_i^TD_i, $$ (no summation on repeated indices) where the $v_i$ are orthonormal column vectors, $D_i$ are diagonal matrices and ...
2
votes
4answers
54 views

Can $AA^T$ be a diagonal matrix for non-square matrix?

Consider a $m \times n$ matrix $A$ where $m \gg n$. Can $AA^T$ be a diagonal matrix (meaning all the diagonal entries are non-zero and the off-diagonal entries are 0).
0
votes
1answer
36 views

Matrix with Limit

Consider the matrix $$A = \begin{bmatrix} \frac{1}{2} &\frac{1}{2} & 0\\ 0& \frac{3}{4} & \frac{1}{4}\\ 0& \frac{1}{4} & \frac{3}{4} \end{bmatrix}$$ What is $\lim_{n→\infty}$$...
0
votes
0answers
8 views

Matrix derivative in images matching problem

Problem Suppose zero-centered matrices $\mathbf{X}$ and $\mathbf{Y}$ of shape $\mathbb{R}^{n\times 2}$. Each row of $\mathbf{X}$ and $\mathbf{Y}$ represents a point on 2-D plane. Therefore, they each ...
2
votes
1answer
40 views

Diagonalization of symmetric matrices

Mas-Collel, Whinston and Green's Microeconomic Theory (3rd edition) asserts that any symmetric matrix can be diagonalized as following: What would be the proof of this?
-2
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1answer
24 views

An n×n matrix is formed using only 0,1,-1 as its elements. Number of such matrices which are skew symmetric is [on hold]

I have no idea how to solve this question .Can anyone please help me out?
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0answers
22 views

Find Jordan norm form of matrix using lambda-matrix techniques

I can't transform this matrix of lambda-matrix techniques. I don't know this method, but I should use this method for solve. Eigenvalues equal to "-1". Guys, help me, please. $$A=\left(\begin{array}{...
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vote
0answers
36 views

Finding representative matrix of quadratic form $ q(x) = x_1x_2 + x_2x_3 + x_1x_3$ with respect to a basis

I'm slightly confused about this question: we have a basis of $R^3$ $B = ${$e_1,e_2, e_3$}, where $x = x_1e_1 + x_2e_2 + x_3e_3$. We need to give the representative matrix of $q$ with respect to ...
1
vote
0answers
29 views

Existence of Rotation Matrix

Given vectors $v_1, v_2 ... v_n \in \mathbb{R}^n$ satisfying $||v_i||_2 = 1$ for all $i$ and $0 \leq\langle\ v_i,v_j\rangle \leq 1$ for all $i, j$. (Not all of them are 0 or 1). Does there always ...
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votes
0answers
11 views

Projection matrix generalized to projection to a surface?

A projection matrix can project a vector into a plane, but is it possible to generalize this such that it can project into a surface instead of a plane?
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0answers
13 views

How to find the vector $(\gamma_1,\gamma_2,\delta)^T$ that characterizes the matrix and product?

I have the matrix: $$H=\left[\begin{matrix}\gamma_1&\delta\\\delta&\gamma_2\end{matrix}\right]$$ and the inner-product: $$<x,y>=x^THy,$$ and has to find the vector $(\gamma_1, \gamma_2, \...
0
votes
0answers
38 views

Is this simple symmetric matrix positive semi-definite?

Let the $n\times n$ symmetric matrix $A$, where $n\geq 9$ be given by \begin{equation} A_{i,j}= \begin{cases} 1.4, &\text{for } 1\leq i=j\leq 9\\ (0.9)^{|i-j|},&\text{for } 1\leq i\neq j\leq 9\...
0
votes
1answer
32 views

Orbit of a symmetric matrix under orthogonal conjugation

Let $A\in M_n(\Bbb{R} )$ be a symmetric matrix. I want to find a general formula for the diagonals of the matrices of the form $g^{-1}Ag$, where $g\in O_n(\Bbb{R})$. Here is what I did : Since $A$ is ...
0
votes
2answers
37 views

Show $N$ is a normal subgroup of $G$ where $G$ is a invertible $2 \times 2$ matrix.

We have $G= \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \text{with $a$ and $c$ in $\{\pm 1\}$ and $b$ in $\mathbb{Z}$} \right\} $, which is given to be a subgroup of the group of ...
0
votes
0answers
21 views

Matrix Equation AB = C with constraints, where both A and B are unknown.

As the title reads, I have the matrix equation: $$ AB = C $$ With the constraint that all elements in B are greater than $0$ and less than $1$, and the last element of each row is $1$. A and B can be ...
-2
votes
0answers
28 views

Determinant of Similar Matrices vs Matrix of Change of Basis

I am little confused when we say that all similar matrices have same determinant. The Proof goes something like: All similar matrices can be represented as $$B=X^{-1}AX$$ hence $det(B) = det(X^{-1}AX) ...
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votes
0answers
11 views

How to find a solution in unknowns in magma? [on hold]

I have a 8x20 matrix M in integers and my vector V is a mix of some integer variables and mostly zeros. I want to solve MX=V, and get the value of X. However, magma doesn't accept variables saying ...
6
votes
0answers
103 views

If $AB=A$, does B have to be the identity matrix?

Suppose $A$ and $B$ are square matrices and that $AB=A$ with $B \neq I$. What does this say about the invertibility of $A$? This question showed up on an exam I took this past spring. I got stuck on ...
1
vote
2answers
21 views

QR - Factorization: If A has full rank then R has non-zeros in the diagonal

$Q$ is an orthogonal matrix. $R$ is an upper triangular matrix. $A \in \mathbb{R}^{m\times n}$ with $m > n$ and its QR-Factorizations is $A = QR$. Show that if $A$ has full rank, then the diagonal ...
1
vote
0answers
17 views

What is the lowest computational complexity of multiplying two non-square matrices?

Based on Wikipedia information, the computational complexity of multiplying two $n\times n$ matrices can be $\mathcal{O}(n^{2.37})$ using algorithms similar to Coppersmith–Winograd. I wonder what if ...
0
votes
0answers
40 views

Number of solutions to binary matrix problem

I want to find the number of possible solution sets for this problem (so that I can compute probabilities later). We want to find two matrices $A_{m \times n} = [a_{i,j}]$ and $B_{m \times n} = [b_{...
2
votes
3answers
28 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 ...
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votes
4answers
46 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
52 views

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

I have the following matrix equation: $\mathbf{x}^\top e^{t\mathbf{A}}\mathbf{y}=c$ $\mathbf{x}$ and $\mathbf{y}$ are vectors of length $k$, $\mathbf{A}$ is a $k\times k$ matrix, and $c$ and $t$ ...
0
votes
1answer
30 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 ...
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votes
0answers
38 views

The matrix exponential [on hold]

Prove that $$\left(e^A\right)^{∗}=e^{A^∗} .$$ I've tried messing around with both sides, I just can't get the two to match up. Any ideas?
0
votes
1answer
37 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
42 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 ...
0
votes
3answers
54 views

Eigenvalues and eigenvectors for the moment of inertia matrix

Find the eigenvalues and eigenvectors for the moment of inertia matrix given by $$I={m\over 2}\left(\begin{matrix} 1 & -1 & 0\\ -1 & 1 & 0\\ 0 & 0 & 2\end{matrix}\right)$$ ...
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votes
0answers
26 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
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3answers
39 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
21 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
17 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}...
0
votes
3answers
22 views

Prove norm of matrix greater than spectral radius

Given an $n \times n$ matrix $A$ with spectral radius $\sigma(A) = 1$, prove that there is an $m_0$ so that $\|A^m\| > 1$ for every $m \geq m_0$. Solution attempt: I think the question is in ...
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
11 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{...
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vote
0answers
34 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') $
3
votes
3answers
39 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).
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votes
0answers
28 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) & \...
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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
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1answer
25 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
43 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 & -...
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votes
0answers
38 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
17 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