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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.

0
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1answer
16 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
1answer
28 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
3answers
38 views

Eigenvalues and eigenvectors for the moment of inertia matrix

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

On determinants and common divisors

Let $n\in\mathbb N$ and let $a_1,\ldots,a_n$ be natural numbers smaller than $10^n$. Write each $a_k$ in base $10$ and add $0$'s to the left of each decimal expansion, if needed, so that each $a_k$ is ...
0
votes
2answers
29 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
12 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
23 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
2answers
1k views

Echelon form of a system of equations?

My prof gave us this definition of an Echelon system: A system of m linear equations in n variables is called an echelon system if m ≤ n. Every variable is the leading variable of at most ...
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votes
0answers
9 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 ...
0
votes
3answers
20 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 ...
0
votes
1answer
861 views

Is it possible to use the deflation algorithm to compute the eigenvalues of a large sparse matrix

I am trying to compute the eigenvalues of a large sparse matrix (about 10% of the values are nonzero). The matrix is real valued, but since it is accumulated by a stochastic process it is not fully ...
5
votes
2answers
1k views

Deriving the inverse of $\mathbf{I}$+Idempotent matrix

Suppose I have an idempotent matrix such that $A^2=A$. From its properties, if $A$ is not an identity matrix, then it is singular. Through trial and error, I can see that for all $I+A$ are invertible. ...
6
votes
0answers
95 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
18 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 ...
0
votes
0answers
31 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
2k views

Operator norm is equal to max eigenvalue

Take a matrix $A \in M_{2 \times 2}(\mathbb{R})$ and consider the norm $\vert\vert A\vert\vert = \sup\limits_{x \in \mathbb{R}^2} \frac{ \vert\vert Ax\vert\vert}{\vert\vert x \vert\vert} = \sup\...
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
0answers
20 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
1answer
22 views

Possible worlds/beliefs/Probability Matrix/Example 3

I the snippet below, copied from the Handbook of Game Theory with Economic Applications, the condition (2.1) says that we can rearrange the columns so that the matrix becomes block diagonal with each ...
1
vote
0answers
16 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
4answers
43 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
3answers
27 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
1answer
64 views

Possible worlds/beliefs/Probability Matrix/Example 2

I have posted a similar question elsewhere, but this one is different. Take the definition and example as below as copied from the Handbook of Game theory with Economic applications. I want to design ...
0
votes
1answer
43 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
1answer
29 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 ...
2
votes
0answers
31 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 ...
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 & -...
3
votes
2answers
91 views

Decomposing a symmetric matrix as a sum of nilpotent matrices

Assume that a real-valued symmetric matrix $M$ with trace zero can be written as $$ M = A + A^T, $$ with $A^2=0$. Given that $M$ is known, how (if possible) can $A$ be found? The diagonal elements ...
0
votes
2answers
40 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
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 ...
-1
votes
0answers
33 views

The matrix exponential [on hold]

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
336 views

Show that subtraction of matrices is neither commutative nor associative. Please review my work.

Show that subtraction of matrices is neither commutative nor associative. My Work Let $A$ and $B$ be $m \times n$ matrices with elements $a_{ij}$ and $b_{i j}$, respectively. Commutativity says ...
0
votes
3answers
38 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
1k views

zeros of $x^*Ax$, a quadratic form

The question hopefully says it all! We have a Hermitian matrix $A=A^* \in \mathbb{C}^n$ and a quadratic form: $f(x)=x^*Ax,~x\in \mathbb{C}^n$ We want to find the solution of $f(x) = x^*Ax = 0$ When ...
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}...
16
votes
2answers
27k views

Outer Product of Two Matrices?

How would I go about calculating the outer product of two matrices of 2 dimensions each? From what I can find, outer product seems to be the product of two vectors, $u$ and the transpose of another ...
0
votes
2answers
56 views

If $A\in\mathbb C^{n\times n}\!$, $\,f\in\mathbb C[t]$ and $f(A)$ diagonalizable, and $f'(A)$ is invertible, then $A$ is diagonalizble.

Suppose that $A$ is a square complex matrix and $f$ is a polynomial in $\mathbb C[t]$ such that $f(A)$ is diagonalizable. If $f'(A)$ is invertible, where $f'$ is the derivative of $f$, prove that $A$ ...
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
2answers
4k views

Construct an example of a 4×4 matrix, with one of its eigenvalues equal to −3, that is not diagonal or invertible, but is diagonalizable

Construct an example of a 4×4 matrix, with one of its eigenvalues equal to −3, that is not diagonal or invertible, but is diagonalizable. I know how to find the eigenvalues, and diagonalizing ...
16
votes
5answers
29k views

For $\det(A)=0$, how do we know if $A$ has no solution or infinitely many solutions?

If the determinant $\det(A)$ of the matrix $A$ of a non-homogeneous system of equations is $0$, then how do we know if it has no solutions or infinitely many solutions? And while we are at it, kindly ...
2
votes
0answers
10 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
33 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') $
6
votes
1answer
45 views
+50

Existence of the $\Omega$ set in “Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization”

In the proof of Lemma 4.3 in [1], they claim the following: Let $U$ be a subspace of $\mathbb{R}^{m\times n}$ with dim$(U)=d$ and let $\delta>0$. Then, there exists a set $\Omega\subset\mathbb{R}^{...
3
votes
3answers
35 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
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
43 views

Efficiently solve a system of equations for only certain degrees of freedom given a known structure

I have an algorithm such that at some point I must solve the following system for $X_5$: $$ \left( \begin{array}{cccccccccc} A_1& B_1& & C_1& & & & & \\ B_7& A_2&...
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= \...
6
votes
0answers
52 views

Is the matrix $\sum_{g \in G} a_g \rho(g)$ normal and what further properties does it have?

Let $\rho$ be the regular representation of $G$. $S \subset G$ a generating set, $|g| := |g|_S=$ word length with respect to $S$. Then I construct such a matrix, where we have some ordering $g_i$ of ...
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 ...