0
votes
0answers
4 views

Canonical form for orthogonal similarity classes

Could someone point me to a reference re canonical forms for classes of matrices in $M_n(\mathbb{C})$ which are unitarily similar? That is, canonical representatives for the equivalence class defined ...
0
votes
0answers
27 views

Updating the LU Factorization

I am looking for a way to update the $LU$ factorization of a general $m \times n$ matrix after adding a column to the matrix. I have to iterate this procedure so I will begin with a matrix that is $m ...
4
votes
0answers
38 views

What is known about the eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular: How are the elements of the ...
0
votes
1answer
41 views

On the decomposition of Stochastic matrices

Let "stochastic" matrix be the matrix whose rows sum to one and deterministic matrix be a stochastic matrix whose all rows consist of a one and zero. For example $\left [ \begin{array}{ccc} 1 & ...
0
votes
0answers
29 views

Matrices with functions as entries

I am interested is studying matrices which have functional entries. Specifically I am looking at quadratic forms of the type $x^T Q(x) x$ where $Q(x)$ is a matrix whose entries are functions of $x$. I ...
0
votes
1answer
16 views

How to test if $m$ vectors are linearly dependent when they are $n$ dimensional and $m < n$

I'll be shocked if this isn't a duplicate, but I haven't had a lot of luck finding an answer to this so far. How do you test if a set of vectors $v_1, \ldots v_m \in \mathbb{R}^n$ are linearly ...
1
vote
0answers
12 views

Birkhoff-Neumann like result for stochastic matrices?

during my research I came along a nice lemma which looks like a Birkhoff-Neumann-theorem result, but in a version for stochastic matrices. Namely, I have: Lemma. Let $M$ be a stochastic matrix, then ...
2
votes
0answers
21 views

Matrix inequality, now what?

I am looking for some motivations. I am reading a book about matrix inequality. e.g. Courant–Fischer equality, eigenvalue of sum of two matrix, ... My question is, so what kind of problems I should ...
0
votes
0answers
20 views

Largest average principal submatrix of a symmetric matrix.

I am wondering if there exists literature on the following problem: Let $X$ be an $n \times n$ symmetric matrix. How do you identify the $k \times k$ principal submatrix of $X$ with the largest ...
4
votes
2answers
34 views

Regular matrices references

Can someone suggest me a book or a lecture note which covers regular matrices with all theories related to it? Any assistance will be much appreciated. (By regular I mean some power of the matrix is ...
5
votes
2answers
72 views

Geometry of Commuting Hermitian Matrices

I am a physicist working on a project dedicated to the quantisation of commuting matrix models. In the appropriate formalism this problem is reduced to a quantisation in a curved space -- the space of ...
3
votes
1answer
41 views

Reference request for positive matrices

I would much appreciate someone suggest me a text book which covers stochastic matrices in depth with all relevant theories.Thanks
2
votes
1answer
53 views

Classification of parabolic elements of a subgroup of $PSL_2(\mathbb R)$

Let $G\subset PSL_2(\mathbb R)$ be the group generated by the matrices $$a_n=\begin{pmatrix} 1 & 2\cot\frac{\pi}{n}\\0 & 1\end{pmatrix},\; c_n = \begin{pmatrix} ...
0
votes
0answers
22 views

Unitary Farey Sequence Matrices

Take the Farey sequence $\mathcal{F}_n$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\frac1{\sqrt{|\mathcal{F}_n|}}\biggr(\exp(2\pi i k a_m)\biggr)_m $$ The dimension of ...
0
votes
1answer
56 views

Matrix representation of complex numbers in exponential form

Do there exist matrices M and P for this equation? Or perhaps M and P dont need to be matrices? I saw this and this question after googling which made me wonder about whether the exponential form of ...
1
vote
0answers
33 views

Matrices of the form $A^p=(a_{ij}^p)$

I am wondering if there is a name for these kind of matrices and if they are interesting or not? Do they even exist? Let $A$ be a $n\times n$ matrix with elements $a_{ij}$. $A= (a_{ij})_{i,j\in\{1, ...
1
vote
0answers
34 views

Finding the $\log$ of a matrix by contour integration

My teacher presented this way of determining the logarithm of a matrix $\Omega$ in class today: $$\log \Omega = \frac1{2\pi i}\oint_{\Gamma} (\zeta I - \Omega)^{-1} \log \zeta \,d\zeta.$$ Does ...
3
votes
2answers
135 views

Smooth spectral decomposition of a matrix

Let $A : x \mapsto A(x)$ be a $C^\infty$ map from the half-plane $\left\{ (x_1,x_2,\cdots,x_n) \in \mathbb{R}^n,\ x_n>0\right\}$ to the space of symmetric matrices with real coefficients. Suppose ...
0
votes
1answer
45 views

Can someone tell me books or papers on subalgebras of $\operatorname{SL}(3)$?

I hope to find the smallest subalgebra of $\operatorname{SL}(3)$ that contain the matrix $$\begin{pmatrix} 0 & a & 0\\0 & 0 & b\\c & d & 0 \end{pmatrix}$$ Are there any ...
6
votes
0answers
156 views

Matrix diagonalization theorems and counterexamples: reference-request.

I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra. In this question I understand the question of matrix diagonalization very broadly: suppose we have ...
0
votes
1answer
40 views

What do real eigenvalues imply for a matrix

Suppose we have a matrix $A \in \mathbb{R}^{n \times n}$ with $\textrm{eig}(A)=\{ \lambda_1, \lambda_2, \ldots, \lambda_n\}$ such that $\lambda_i \in \mathbb{R}$. Does the realness of the eigenvalues ...
3
votes
2answers
52 views

Proof that diagonally dominant matrices are regular - Reference request

I know that it is easy to proof that diagonally dominant matrices are regular (non-singular) by the gershgorin circle theorem. But the theorem that diagonally dominant matrices are regular was ...
2
votes
2answers
76 views

Expressions for Permanent of a Matrix

Given that the permanent of a matrix can be written in a similar form as the determinant, as a sum of permutations of the elements of the matrix, is there also a relationship between the permanent and ...
2
votes
1answer
109 views

Matrix Theory book Recommendations

I'm currently reading Sheldon Axler's "Linear Algebra Done Right". Can anyone recommend any good books on matrix theory at about the same level that might compliment it?
4
votes
0answers
93 views

Can the “inducing” vector norm be deduced or “recovered” from an induced norm?

Can the "inducing" vector norm be deduced or "recovered" from an induced (operator) norm? This question occurred to me after seeing this question. I'm hoping that perhaps there exists something like ...
2
votes
0answers
19 views

Inequalities on Matrix Minimax?

Suppose I have a matrix $M$. How can I get a good bound on the minimax quantity $$ \min_{i}\max_{j}M_{ij} $$ or variations thereof? Links to literature would be greatly appreciated.
0
votes
0answers
149 views

The ith leading principal submatrix obtained by interchanging columns

Let $S \in GL_n(\mathbb{Q})$ be a non-singular symmetric $n\times n$ matrix with LDU decomposition. Let $L^T=U=(f_{ij})$ where $f_{ii}=1$ for all $i = 1, \ldots, n$ and $f_{ij} = 0$ if $ i > j$, ...
1
vote
0answers
36 views

Is there a special name for matrices with $M[j,i] = M[i,i] - k, i \neq j$?

Backgroud: I am working on a computer science problem and arrives at a matrix $M$ with the following property: The size of Matrix $M$ is $n\times n$. For each row $j$, we have $M[j,i] = ...
4
votes
1answer
134 views

What are the one-parameter subgroups of GL?

Are the multiplicative one-parameter subgroups of the general linear group (i.e., morphisms $\lambda:\Bbbk^\times\to\mathrm{GL}_n\Bbbk$ of algebraic groups) completely classified? The obvious ...
0
votes
4answers
110 views

Book Searching in Stability Theory.

Can anyone recommend me a book on Stability Theory with an intuitive approach? I have some course notes on that subject, but it's really abstract and theoretical. I really want to understand it, ex: ...
3
votes
2answers
1k views

Best Books to learn Proof-Based Linear Algebra and Matrices

So I'm in a really serious problem. It's my first year at university and I'm doing a CS major. The math is already getting serious and I'm lost, really lost. It's all about matrices so far and the ...
0
votes
1answer
75 views

Fixed point of matrix

Suppose that $a$ is a fixed point of matrix $A$, what that means? What is a fixed point of matrix? Thank you!
2
votes
1answer
93 views

Study of Matrix Calculus

I need to study matrix calculus such as integration, differentiation, differentiation of functions of determinants and inverse matrices and then also other matrix based calculations such as ...
1
vote
2answers
71 views

Why does $\ln(I+A)$ converge when $\|A\|<1$?

Suppose $A$ is an $n\times n$ matrix with real entries. Often times you can take the logarithm $$\ln(I_n+A)=A-A^2/2+A^3/3-\cdots+(-1)^{n+1}A^n/n$$ when the matrix norm $\|A\|<1$. Is there a ...
6
votes
1answer
140 views

Integrals of matrix functions

I've stumbled across some math I've never really encountered before, and I would love it if someone could provide me with some useful references and texts on it. I'm dealing with integration over the ...
4
votes
1answer
91 views

Matrix-product-integrals?

Whereas the conventional "sum integral" is $$ \lim_{\Delta x\to 0} \sum_i f(x_i)\,\Delta x, $$ a "product integral" is $$ \lim_{\Delta x\to 0} \prod_i f(x_i)^{\Delta x}. $$ Now you're thinking: just ...
1
vote
0answers
46 views

Textbook on the Theory of Orthogonal Matrices

Is there a textbook with good coverage on the theory of orthogonal matrices? If possible, I would prefer that the orthogonal matrices are allowed to have complex entries. Note that I am not ...
1
vote
0answers
68 views

Matrix functions via Cauchy integral

I shall be much obliged if one provides me with references on calculation of "standard" matrix functions by use of Cauchy integral, such as matrix exponent matrix logarithm matrix square root matrix ...
0
votes
2answers
148 views

ISO brief primer on special matrices

I am looking for a brief primer on the following types of matrices: stochastic, doubly stochastic, symplectic, Vandermonde, Hadamard, permutation, tridiagonal and circulant. Nothing too deep, just a ...
4
votes
1answer
480 views

Cholesky for Non-Positive Definite Matrices

I am trying to approximate a NPD matrix with the nearest PD form and compute its Cholesky decomposition. I know that the usual method is to perform an eigenvalue decomposition, zero out the negative ...
3
votes
0answers
74 views

The intuition behind a matrix of a Hamiltonian?

We have derived an elegant partition function for a problem which resembles a quantized model taking the particles to be Bosons. The related Hamiltonian for every $i$th ensemble is there: ...
1
vote
1answer
688 views

What is matrix reduction to normal form PAQ?

Here is my university syllabus. I started doing math in vacation just to get a head start because I am a dunce in math. So, I began with chapter 2 - matrices - because it looked easier. I went half ...
0
votes
1answer
133 views

A matrix has a real logarithm if it has a positive spectrum.

The title is a proposition I read in my notes that's left with no proof. Where can I read one?
0
votes
1answer
85 views

Definition of $\exp(A)$ in terms of spectral decomposition.

I am read this question Plugging a matrix multiplied by an imaginary number in the exponential function. Here the question'author defined $$\exp(A) := \sum_{1\le k\le n}\exp(\lambda_k) P_k$$ What ...
1
vote
3answers
409 views

A constrained linear least Frobenius norm problem:$\min_{X} \|A-XB\|_F$ subject to $Xv=0$?

Assume we are given two matrices $A, B \in \mathbb R^{n \times m}$ and a vector $v \in \mathbb R^n$. $\|\cdot\|_F$ is the Frobenius norm of a matrix. How can we solve $$\min_{X \in \mathbb R^{n ...
6
votes
3answers
249 views

An “Itzykson-Zuber”-like integral

I been told that there exists an integration formula, which states (or something of this sort) $$ \int_{U(N)} dU \det[(\mathbb I+XUYU^{-1})^{-r}\propto ...
1
vote
1answer
192 views

Book about applied linear algebra to the 3D world?

I have no problem understanding the basics of the linear algebra, but I also know that the linear algebra it's still an analytical approach to real world scenarios that can be solved with matrices and ...
2
votes
1answer
127 views

Construction of Hadamard Matrices of Order $n!$

I'm trying to get a hand on Hadamard matrices of order $n!$, with $n>3$. Payley's construction says that there is a Hadamard matrix for $q+1$, with $q$ being a prime power. Since $$ n!-1 \bmod 4 = ...
1
vote
1answer
96 views

Good source for self study of matrix decompositions

What is a good source for study of various types of matrix decomposition, which is both comprehensive and also includes applications? It should at least cover LU, RQ, SVD, spectral, Schur, and ...
11
votes
1answer
1k views

The Determinant of a Sum of Matrices

Given $N$ $n \times n$ matrices $\mathsf{A}^{1}, \dots, \mathsf{A}^{N}$, \begin{align} \det \left( \sum_{i = 1}^{N} \mathsf{A}^{i} \right) = \sum_{\sigma \in S} \det \mathsf{A}^{\sigma}, \end{align} ...