Tagged Questions

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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3
votes
2answers
581 views

Dimension of $GL(n, \mathbb{R})$

Why is the group $GL(n, \mathbb{R})$ of dimension $n^{2}$?
3
votes
1answer
1k views

Finding determinant of an infinite matrix

I am trying to find the simplest way to get an expression of the determinant of the following infinite matrix as m tends to infinity. $$ \left[\begin{array}{cccccc} 1 & a_{1} & 0 & ...
3
votes
1answer
546 views

Irreducible Representations of Matrix Algebras

I am currently reading the book Spin Geometry by Lawson/Michelsohn to understand Dirac Operators and related topics. At some point it uses representation theory to classify Clifford Algebras. In ...
1
vote
0answers
128 views

Spectral decomposition of the matrix ODE

I have a matrix ODE $\dot X = f(X,t)$ on the symmetric matrix with the initial condition $X(0) = X_0$, where $X_0$ is the positive definite and symmetric. Then by the spectral decomposition $X_0 = ...
3
votes
1answer
486 views

The square root of positive definite matrix

Let $M$ be the manifold of real positive definite $n \times n$ matrices, define a mapping $i:A \to \sqrt A$ (where $A\in M$ and $\sqrt A$ means the unique positive definite square root of $A$). Please ...
2
votes
3answers
1k views

(good) numerical inversion of an almost singular matrix: is it possible?

Ok, so I know that if I have a system that looks like Ax=b it is foolish to solve it by solving for the inverse of A and one should instead use something like ...
2
votes
1answer
96 views

Why is this matrix positive?

I run into the following fact: Let $X$ be $n \times n$ positive semidefinite matrix, and let $a$ be an $n$ dimensional vector consisting of diagonal entries of $X$ (i.e. $a_i = X_{ii}$). Suppose ...
1
vote
1answer
33 views

what should be the dimensions of the matrix mentioned in this question

what should be the dimension of matrix to be multiplied with a $4\times 1$ matrix so that i can get a $2 \times 1$ matrix ? To make it clear: $\left(\fbox{?}\times\fbox{?}\right)\left(4\times ...
1
vote
1answer
904 views

Matrix Transformation Onto?

A linear transformation $T\colon\mathbb{R}^3\to\mathbb{R}^2$ whose matrix is $$\left(\begin{array}{ccc} 1 & 3 & 3\\ 2 & 6 & -3.5+k \end{array}\right)$$ is onto if and only if ...
7
votes
2answers
1k views

Can a basis for a vector space be made up of matrices instead of vectors?

I'm sorry if this is a silly question. I'm new to the notion of bases and all the examples I've dealt with before have involved sets of vectors containing real numbers. This has led me to assume that ...
0
votes
0answers
49 views

Maintaining the line with the 2D iterands

Suppose a linear system is given $$AX=B,$$ where $A\in\mathbb{R}^{n\times n}$ is a symmetric strictly diagonal matrix, and $X, B\in\mathbb{R}^{n\times 2}$. Therefore, the 2D Jacobi iterative solver is ...
2
votes
1answer
179 views

Natural number matrix solutions to $\sigma_i\sigma_j+\sigma_j\sigma_i = I\delta_{ij}$

Given the two matrices: $\sigma_i$ and $\sigma_j$ we can construct a Clifford algebra based on the anti commutator rule: $$\{\sigma_i,\sigma_j\}=\delta_{ij}1$$ where $\delta_{ij}$ is the Kronecker ...
2
votes
1answer
51 views

The form of the states on an algebra of $n\times n$ matrices with complex entries

I have a question concerning $n \times n$ matrices. Denote by $M_n(\mathbb{C})$ the algebra of all $n \times n$ matrices with complex entries. Let $\phi$ be a state on $M_n(\mathbb{C})$ i.e. a linear ...
0
votes
1answer
1k views

Some theorem about block matrix determinants with symmetric inner matrices?

I could do this problem with bruteforce but I think there must be some elegant theorem that helps to calculate the determinant with the block matrix (here having symmetric matrices inside) such as: ...
0
votes
1answer
216 views

Let $A$ be a square matrix such that $A^2 = A$. Show that $A$ cannot be a strictly diagonally dominated matrix unless A is the identity matrix.

Let $A$ be a square matrix such that $A^2 = A$. Any idea how to show that $A$ cannot be a strictly diagonally dominated matrix unless $A$ is the identity matrix.
1
vote
2answers
1k views

Determinant of Large Matrix with Gauss rule?

$$A=\begin{pmatrix} 1 & -1 & 0 & 2 \\ 2 & 1 & 0 & 0 \\ 1 & 1 & 2 & 2 \\ 0 & 0 & 1 & 1 \\ \end{pmatrix}$$ With the lower determinant method, I ...
3
votes
1answer
59 views

Properties of matrices

Have I got these properties of matrices correct?
2
votes
2answers
97 views

Help to evaluate determinant

I want to evaluate the determinant of the $n \times n$ matrix $\left|\begin{array}{ccccc} 1 & 0 & \ldots & 0& 0 \\ 0 & 0 & \ldots & 0 & -a\\ 0 & 0 & \ldots ...
1
vote
0answers
108 views

Matrix Factorization problem

Okay so here is what I am trying to do, I have a matrix $X$ consisting of values $X_{nm}$. I have to find vectors $P_n$ and $Q_m$ of length $k$, such that the value $$Y_{nm} = P_n \text{ . } ...
2
votes
1answer
222 views

Similarity Transformation

Let $G$ be a subgroup of $\mathrm{GL}(n,\mathbb{F})$. Denote by $G^T$ the set of transposes of all elements in $G$. Can we always find an $M\in \mathrm{GL}(n,\mathbb{F})$ such that $A\mapsto ...
1
vote
1answer
626 views

Find 3D rotation vector and angle to transform a rectangle into a given quadrilateral

I have a given rectangle that I need to transform into a given quadrilateral shape that resulted from a rotation and translation in 3D space, and subsequent projection. ...
2
votes
2answers
259 views

Measure to compare matrix $A$ and permuted matrix $B$

I have a matrix $A$ and matrix $B$ of same dimension. We generally use $||A-B||_F^2$ (Forbenius norm) to compare these two matrix how close they are to each other. Here we assume the $col_i$ of matrix ...
3
votes
2answers
506 views

Conjugacy classes in a matrix group

Consider the matrix group $PGL_{2}(\mathbb{F}_{q})$ for $q$ odd. Why is it that $\begin{pmatrix} -1 & 0\\ 0 & 1\end{pmatrix}$ has $q(q + 1)/2$ elements in its conjugacy class while ...
1
vote
0answers
468 views

Efficient Cholesky decomposition of inverse matrix

I want to generate random numbers from a multivariate normal distribution in Matlab. Normally, this is done like: $w = \overline{w} + \text{chol}(\Sigma) \cdot \vec{l}$ But in my case I don't know ...
5
votes
0answers
228 views

Finding conjugacy classes of $PGL_{2}(\mathbb{F}_{q})$

Assume $q$ is odd. How does one go about finding the conjugacy classes of $PGL_{2}(\mathbb{F}_{q})$? I know that for $GL_{2}(\mathbb{F}_{q})$, one can consider the possible Jordan Normal Forms of the ...
1
vote
3answers
125 views

Question about norms of a matrix when exchanging two of its rows

Assume I exchange two rows of a square complex $n\times n $ matrix. Are the Euclidean norm and the Hilbert-Schmidt norm of the new matrix (obtained from the first one by exchanging two of its rows) ...
5
votes
1answer
442 views

Hilbert Schmidt Norm-Rank-inequality

Problem: Let $A_{n.n}$ be square complex matrix. Prove the following: $$\left \| A \right \|=\left \| A \right \|_{HS}\Leftrightarrow rank(A)\leqslant 1$$. Where $\left \| . \right \|_{HS} $ is the ...
0
votes
1answer
87 views

Differential equation for a matrix-valued function

Let $Y\left(t\right)$ be a matrix function, then does $\frac{d}{dt}Y^{T}Y=O$ necessarily imply $Y^{T}\left(t\right)Y\left(t\right)\equiv I$ ? Why or why not? Here $Y^{T}$ the transpose of $Y$ , $O$ ...
2
votes
2answers
2k views

How do I calculate the derivative of matrix?

I'd like to expand a real, symmetric and positive definite Matrix $M$ into a Taylor series. I'm trying to do $$ M (T) = M(T_0) + \frac{\partial M}{\partial T} (T-T_0) + \cdots$$ $T$ is a algebraic ...
1
vote
2answers
1k views

Inequality involving norm of matrix integral

This question seems basic but I could not find an answer. I have seen the inequality $$\left\|\int_a^b x(t) dt \right\| \leq \int_a^b \left\| x(t) \right\| dt $$ where $x(t) \in \mathbb{R}^n$ is a ...
2
votes
2answers
86 views

Finding a set of similarity representatives for a collection of $2\times 2$ complex matrices

Here is the problem statement: Find a subset $Y$ of $X:=\{A \in \text{Mat}_{2\times 2}(\mathbb{C})\ |\ A^4=A\}$ so that the following occur: If $A$ $\in$ $X$ then $\exists$ $B$ $\in$ $Y$ such that ...
3
votes
1answer
252 views

Note on Ring Homomorphisms of Matrices Rings

Assume that $\mathbb{F}$ is a field, and let $\mathbb{M}_{t}\left( \mathbb{F}\right) $ be the ring of matrices of order $t$ over $\mathbb{F}$. Does there exist a non-trivial ring homomorphism from ...
1
vote
1answer
179 views

Proof on the inequality involving matrix splitting and trace operator

Suppose positive definite matrices $V, B, D\in\mathbb{R}^{n\times n}$ are given, where $D$ only contains diagonal entries of $V$, i.e., $D=diag(V)$, and $X, G\in\mathbb{R}^{n\times 2}$. Could the ...
0
votes
1answer
950 views

Hermitian matrices that commute

My question is: If $A$ and $B$ are two Hermitian matrices, and $AB$ is also a Hermitian matrix, then how do prove that both $A$ and $B$ are diagonalizable through the same unitary matrix (i.e the ...
2
votes
1answer
192 views

Is there a closed-form solution to this linear algebra problem?

$A$ and $B$ are non-negative symmetric matrices, whose entries sum to 1.0. Each of these matrices has $\frac{N^2-N}{2}+N-1$ degrees of freedom. $D$ is the diagonal matrix defined as follows (in ...
10
votes
3answers
4k views

Get Transformation Matrix from Points

I have built a little C# application that allows visualization of perpective transformations with a matrix, in 2D XYW space. Now I would like to be able to calculate the matrix from the four corners ...
0
votes
1answer
248 views

Invertible complex square matrix

Here is a small question: I was reading a problem in a textbook where the question is: Prove that $A$ is invertible? (where $A$ is a complex square matrix). In the solution: the author proved that ...
2
votes
1answer
2k views

How to find the orthonormal transformation that will rotate a vector to the x axis?

I am having trouble remembering linear algebra. I need to find the orthonormal transformation that will rotate a 3-dimensional vector to the x axis. I could not find any similar question on the net. ...
0
votes
1answer
160 views

Chain Rule and Homogenous Coordinates

I have a vector $\tilde{p} = (x,y,z)$ (homogenous coordinates). The corresponding non-homogenous vector is $p = (x/z, y/z)$. Now the $\tilde{p}$ is a result of some linear transform $R(\theta)$ of ...
1
vote
1answer
286 views

Determinant of a random matrix

Given the set $A=\{0,1\}$ of all the real numbers between $0$ and $1$, we can build the square random matrix: $$H_2=\begin{bmatrix}h_{11} & h_{12} \\ h_{21} & h_{22}\end{bmatrix}$$ where the ...
7
votes
4answers
604 views

Cool/Useful Examples of Characteristic and Minimal Polynomials?

I'm teaching a Linear Algebra II undergrad course and for the section on characteristic & minimal polynomials, I really don't want to just give the students a bunch of matrices that have no ...
3
votes
1answer
585 views

Hermitian positive semi-definite-Square root

Problem: Let $A$ be a Hermitian positive semi-definite $n$ by $n$ matrix (The field of scalars is $\mathbb{C}$). Let $B$ be an $n$ by $n$ matrix that commutes with A. Prove that $B$ and $\sqrt{A}$ ...
8
votes
2answers
8k views

What is the proof that covariance matrices are always semi-definite?

Suppose that we have two different discreet signal vectors of $N^{th}$ dimension, namely $\textbf{x}[i]$ and $\textbf{y}[i]$, each one having a total of $M$ set of samples/vectors. $\textbf{x}[m] = ...
2
votes
1answer
643 views

How to diagonalize infinite symmetric banded matrices?

Given the tridiagonal symmetric infinite matrix of 0 and 1's $$ \left( \begin{matrix} 0&1&0&0&\ldots&0\\ 1&0&1&0&\ldots&0\\ ...
3
votes
2answers
41 views

limit law and product of matrices

Are there two $ n\times n $ matrices $A$ and $B$ such that $ \lim_{m\to\infty} A^m$ and $ \lim_{m\to\infty} B^m $ both exists but $ \lim_{m\to\infty} (A \cdot B)^m $ doesn't ?
1
vote
1answer
384 views

Transitive reduction: calculating “relation composition” of matrices?

I have graphs represented by matrices. For example, $\begin{matrix} 0&0&0\\1&0&0\\1&1&0\end{matrix}$ Produces this graph: The graphs are supposed to be transitive, i.e. ...
1
vote
2answers
147 views

$A = LDL^T \Rightarrow $all of the main diagonal entries of $D$ are positive?

$A$ is symmetric positive definite and $A = LDL^T$, where $L$ is unit lower triangular and $D$ is diagonal. I want to prove that the main-diagonal entries of $D$ are all positive. I have tried ...
1
vote
2answers
427 views

Computing the linear transformations, dimensions and change of basis.

Have the following let $P_2(R)$ denote a vector space of the real polynomial functions of degree less than or equal to two and let $B:=[P_0,P_1,P_2]$ denote the natural ordered basis for $P_2(R)$ (so ...
2
votes
0answers
70 views

Random matrix with non-identical variances

Consider $A$ a $n \times n$ random matrix with centered Gaussian entries $A_{i,j}$ such that $$\mathbb{E}[A_{i,j}^2]=\sigma_j^2/n$$ The variances depend on the column only. What do we know on the ...
0
votes
0answers
100 views

Convert triangular real matrix to hermitian

We are developing some computer program which at some point uses a library (for which we do not have access to its source code) to solve the general eigenvalue problem; given two input real symmetric ...