# Tagged Questions

Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to [tag:linear-algebra]). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur ...

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### On the positivity of matrices

For given $n$, the following $n\times n$ real matrix $M=M^{T}$ is called positive, if $x^{T}M x\geq 0$ holds for all non-negative real $x_1,x_2,\cdots,x_n$, where $x=(x_1,x_2,\cdots,x_n)^T$. ...
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### A generalization of van der Waerden's conjecture

I am wondering if the following generalization of van der Waerden's conjecture is true. Suppose A is an n x n non-negative matrix with all column sums equal to 1, and the sum of row i equal to $T_i$. ...
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### when does elementwise-log preserve positive-semidefiniteness?

Let $Z$ be a positive semidefinite matrix with nonnegative entries, and define $X=\log(1+Z)$, where the $\log$ is taken entrywise, i.e., $X_{ij}=\log(1+Z_{ij})$. Are there some simple sufficient ...
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### Optimization of a function of a positive definite matrix and its inverse

This question is a little ill-posed, but I've been playing with some equations and am just wondering if this resembles any known problems that have been solved. Suppose I have two real, positive ...
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### Ergodic theory reference for converging sequences of matrices

I have been told that the following is a well known theorem in ergodic theory & have been given the book by Furstenberg as a reference. However, I cannot find such a statement in it. Would anyone ...
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Let $A$ be a real $n\times n$ matrix and let $\mu_1, \dots, \mu_n$ the (generalized, complex) eigenvalues of $A$. Assume that $$0 < \alpha < \mathrm{Re}(\mu_1) < \dots < \mathrm{Re}(\mu_n)... 2answers 562 views ### Matrix groups and presentation Suppose K is a number field and I have a subgroup of GL_2(K) for which I know a (finite) set of generators. Is there an algorithm that gives me a presentation of the group? More precisely, the ... 1answer 111 views ### Calculating the dimension of the algebra generated by some given matrices Let X_1, X_2, \ldots, X_d be n \times n matrices over some field K. I want to calculate the dimension of the unital algebra generated by X_1, X_2, \ldots, X_d for some examples in a problem I ... 0answers 2k views ### Matrix Transpose Similarility A famous problem in linear algebra is that "A n\times n matrix A over a field \mathbb{F} is similar to its transpose A^T." I know one proof using Smith Normal form. However, I want to know an ... 2answers 564 views ### matrices whose entries sum to zero Let A be a non-singular matrix and let s(A) be the sum of its entries. Under which conditions can it be assured that s(A) \neq 0? if you like, you can assume that A is symmetric. Here is an ... 2answers 437 views ### Is Ryser's conjecture on permanent minimizers still open? Let A(k,n) be the set of \{0,1\} matrices of order n with all their line sums equal to k. Conjecture number 5 on the list from Minc's book, attributed to Ryser, says that if A(k,n) ... 2answers 330 views ### A short question about the DFT matrix Is the DFT matrix the unique* unitary matrix with all entries of same magnitude? (*up to some trivial transformations) 2answers 323 views ### On a determinant inequality of positive definite matrices Assume that B and A are two positive definite matrices. Take B^* a block diagonal matrix with block B_{11} and B_{22} of B. This means the following:$$ B=\left[\begin{array}{ll} B_{11}&...
Is the $S$-matrix conjecture still open? I mean the one listed as Problem 7 in this survey.