# Tagged Questions

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### Is there anything “nice” about the set of normal matrices (over $\Bbb R$ and $\Bbb C$?)

Normal matrices are of course useful to any linear algebra buff, not least because of the spectral theorem. However, taken as a whole, they tend to have some not-so-nice properties. For example: ...
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### Proof of Minkowski determinant inequality

I wonder where can I find the proof for the Minkowski determinant inequality? ( i.e., given two positive definite n x n symmetric matricies A and B, $det(A+B)^{1/n}\ge det(A)^{1/n}+det(B)^{1/n}$ ) ...
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### What is a good reference for learning about induced norms?

Wikipedia tells me a little about it. Following the wiki-link treasure hunt leads me to topics such as "p-norms on finite dimensional vector spaces". Which makes me want to ask: what's a good ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Matrix inequality, now what?

I am looking for some motivations. I am reading a book about matrix inequality. e.g. Courantâ€“Fischer equality, eigenvalue of sum of two matrix, ... My question is, so what kind of problems I should ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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Let $G\subset PSL_2(\mathbb R)$ be the group generated by the matrices $$a_n=\begin{pmatrix} 1 & 2\cot\frac{\pi}{n}\\0 & 1\end{pmatrix},\; c_n = \begin{pmatrix} ... 0answers 22 views ### Unitary Farey Sequence Matrices Take the Farey sequence \mathcal{F}_n with values a_m\in \mathcal{F}_n and put them into a vector$$ \vec v_k=\frac1{\sqrt{|\mathcal{F}_n|}}\biggr(\exp(2\pi i k a_m)\biggr)_m $$The dimension of ... 1answer 57 views ### Matrix representation of complex numbers in exponential form Do there exist matrices M and P for this equation? Or perhaps M and P dont need to be matrices? I saw this and this question after googling which made me wonder about whether the exponential form of ... 0answers 33 views ### Matrices of the form A^p=(a_{ij}^p) I am wondering if there is a name for these kind of matrices and if they are interesting or not? Do they even exist? Let A be a n\times n matrix with elements a_{ij}. A= (a_{ij})_{i,j\in\{1, ... 0answers 34 views ### Finding the \log of a matrix by contour integration My teacher presented this way of determining the logarithm of a matrix \Omega in class today:$$\log \Omega = \frac1{2\pi i}\oint_{\Gamma} (\zeta I - \Omega)^{-1} \log \zeta \,d\zeta.$$Does ... 2answers 139 views ### Smooth spectral decomposition of a matrix Let A : x \mapsto A(x) be a C^\infty map from the half-plane \left\{ (x_1,x_2,\cdots,x_n) \in \mathbb{R}^n,\ x_n>0\right\} to the space of symmetric matrices with real coefficients. Suppose ... 1answer 46 views ### Can someone tell me books or papers on subalgebras of \operatorname{SL}(3)? I hope to find the smallest subalgebra of \operatorname{SL}(3) that contain the matrix$$\begin{pmatrix} 0 & a & 0\\0 & 0 & b\\c & d & 0 \end{pmatrix}$$Are there any ... 0answers 161 views ### Matrix diagonalization theorems and counterexamples: reference-request. I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra. In this question I understand the question of matrix diagonalization very broadly: suppose we have ... 1answer 42 views ### What do real eigenvalues imply for a matrix Suppose we have a matrix A \in \mathbb{R}^{n \times n} with \textrm{eig}(A)=\{ \lambda_1, \lambda_2, \ldots, \lambda_n\} such that \lambda_i \in \mathbb{R}. Does the realness of the eigenvalues ... 2answers 53 views ### Proof that diagonally dominant matrices are regular - Reference request I know that it is easy to proof that diagonally dominant matrices are regular (non-singular) by the gershgorin circle theorem. But the theorem that diagonally dominant matrices are regular was ... 2answers 79 views ### Expressions for Permanent of a Matrix Given that the permanent of a matrix can be written in a similar form as the determinant, as a sum of permutations of the elements of the matrix, is there also a relationship between the permanent and ... 1answer 112 views ### Matrix Theory book Recommendations I'm currently reading Sheldon Axler's "Linear Algebra Done Right". Can anyone recommend any good books on matrix theory at about the same level that might compliment it? 0answers 96 views ### Can the “inducing” vector norm be deduced or “recovered” from an induced norm? Can the "inducing" vector norm be deduced or "recovered" from an induced (operator) norm? This question occurred to me after seeing this question. I'm hoping that perhaps there exists something like ... 0answers 20 views ### Inequalities on Matrix Minimax? Suppose I have a matrix M. How can I get a good bound on the minimax quantity$$ \min_{i}\max_{j}M_{ij} $$or variations thereof? Links to literature would be greatly appreciated. 0answers 161 views ### The ith leading principal submatrix obtained by interchanging columns Let S \in GL_n(\mathbb{Q}) be a non-singular symmetric n\times n matrix with LDU decomposition. Let L^T=U=(f_{ij}) where f_{ii}=1 for all i = 1, \ldots, n and f_{ij} = 0 if  i > j, ... 0answers 36 views ### Is there a special name for matrices with M[j,i] = M[i,i] - k, i \neq j? Backgroud: I am working on a computer science problem and arrives at a matrix M with the following property: The size of Matrix M is n\times n. For each row j, we have M[j,i] = ... 1answer 136 views ### What are the one-parameter subgroups of GL? Are the multiplicative one-parameter subgroups of the general linear group (i.e., morphisms \lambda:\Bbbk^\times\to\mathrm{GL}_n\Bbbk of algebraic groups) completely classified? The obvious ... 4answers 110 views ### Book Searching in Stability Theory. Can anyone recommend me a book on Stability Theory with an intuitive approach? I have some course notes on that subject, but it's really abstract and theoretical. I really want to understand it, ex: ... 2answers 1k views ### Best Books to learn Proof-Based Linear Algebra and Matrices So I'm in a really serious problem. It's my first year at university and I'm doing a CS major. The math is already getting serious and I'm lost, really lost. It's all about matrices so far and the ... 1answer 76 views ### Fixed point of matrix Suppose that a is a fixed point of matrix A, what that means? What is a fixed point of matrix? Thank you! 1answer 95 views ### Study of Matrix Calculus I need to study matrix calculus such as integration, differentiation, differentiation of functions of determinants and inverse matrices and then also other matrix based calculations such as ... 2answers 71 views ### Why does \ln(I+A) converge when \|A\|<1? Suppose A is an n\times n matrix with real entries. Often times you can take the logarithm$$\ln(I_n+A)=A-A^2/2+A^3/3-\cdots+(-1)^{n+1}A^n/n$$when the matrix norm \|A\|<1. Is there a ... 1answer 141 views ### Integrals of matrix functions I've stumbled across some math I've never really encountered before, and I would love it if someone could provide me with some useful references and texts on it. I'm dealing with integration over the ... 1answer 91 views ### Matrix-product-integrals? Whereas the conventional "sum integral" is$$ \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 ... 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 ... 0answers 69 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 ... 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 ... 1answer 509 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 ... 0answers 75 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 ith ensemble is there: ... 1answer 740 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 ... 1answer 134 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? 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$$What ... 3answers 431 views ### A constrained linear least Frobenius norm problem:\min_{X} \|A-XB\|_F subject to Xv=0? Assume we are given two matrices A, B \in \mathbb R^{n \times m} and a vector v \in \mathbb R^n. \|\cdot\|_F is the Frobenius norm of a matrix. How can we solve$$\min_{X \in \mathbb R^{n ...
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 ...