# Tagged Questions

Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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### Wedge product of entries of a matrix & Volume form of the Siegel metric

Let $A=(a_{ij})$ be an $n\times n$ square matrix, and $\omega(A)=\bigwedge\limits_{i,j=1}^na_{ij}$ be the wedge product of its entries. Then, if $B=UA=(b_{ij})$ for some square matrix $U$, I think one ...
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### proof Hadamard's Inequality [on hold]

Theorem 4.2. Hadamard’s Inequality. Suppose A is positive semideﬁnite of size n. Then |A|≤ [A]11···[A]nn. Proof. Let A be any positive semideﬁnite matrix of size n. Note that In is a postive ...
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### Whether a given algebra is the algebra of endomorphisms for a vector space

Let $\mathbb{F}$ be a field and let $A$ be an associative unital $\mathbb{F}$-algebra. Is there a criterion to let me know if $A$ is isomorphic to the algebra $\mbox{End}(\mathbf{V})$ of endomorphisms ...
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### Need help with paper written in Russian… Yorgov's paper on self-dual codes with automorphisms of odd order

I came across this paper by V.I.Yorgov named "Binary self-dual codes with automorphisms of odd order". (Actually I was first reading another paper by Borello that cited this paper. It later become ...
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### Complexity for solving linear equations?

What is the best known complexity for finding a vector $x \in \mathbb{R}^n$ to minimize $||Ax - b||^2$ and/or to solve (when possible) the system of linear equations $Ax=b$? I am interested in ...
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### Expected amount of linearly dependent random vectors? [closed]

Given a random Matrix $A\in \mathbb{F}_2^{n\times n}$ what is the expectation value of the amount of linearly dependent row-vectors of $A$? EDIT: As said in the comments, I'm looking for the ...
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### How are eigenvalues and eigenvectors affected by adding the all-ones matrix?

Given an $n \times n$ matrix $A$ and the $n\times n$ all-ones matrix $J = (1)_{ij}$, I'm interested in the relation between the eigenvalues and eigenvectors of the matrices $A$ and $A+J$, or more ...
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### showing that a matrix has repetitive values?

Here my primary aim is to calculate the stationary distribution of a DTMC using left-eigen values i.e, $\pi = \pi*P$. But for some matrices, I observe that some states a same stationary probability. ...
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### An extremal problem on matrices

Is it possible to determine (or give bounds for) the following extremal problem: Let $k,m,r$ be positive integers such that $k,m \geq r$. What is the least number $n$ such that for any $r \times n$ ...
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### Rotating a known vector over two axis-es to result to another known vector [closed]

Lets assume i have a known vector, for example x = [1,0,0] After 2 rotations, one over the y axis and one over the z axis, i result in a vector which in this example is x' = [0.5774, 0.5774, 0.5774] ...
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### Linear algebra in terms of abstract nonsense?

The categories of vector spaces and finite dimensional vector spaces are pretty much as nice as can be, I think. I was wondering what portions of basic linear algebra (first couple of courses) fall ...
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### distance from the mean of a normal distribution to the span of a random sample

Let $W$ be a $d\times k$ matrix whose columns are sampled from a multivariate normal distribution with mean $\mu$ and unit covariance. I'm interested in $|\mu - WW^+\mu|$, that is the distance from ...
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### A question on linear groups

Asking this question I have made a mistake joining my main question with two simple ones, so it hasn't received enough attention, however there was a partial answer, which was not elaborated, and now ...
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### How does grade projection act on homogeneous multivectors in geometric algebra?

I'm reading Clifford Algebra to Geometric Calculus by Hestenes, and struggling with an early result about reversion inside of a grade-projection operator. It is noted that $A_r$ and $B_s$ are ...
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### Image of composition of integral upper triangular matrices

For $A,B$ integral upper triangular matrices on $\mathbb{Z}^k$, do we know something about the image $\text{im}(AB)$ in terms of $\text{im}(A)$, $\text{im}(B)$, unions, intersections, determinants, ...
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### Bijection modeling isomorphism of infinite-dimensional vector spaces

Let $T : V \to W$ be an isomorphism of vector spaces with bases $B_V$ and $B_W$, which may be of any cardinality. Does there exist a bijection $f : B_V \to B_W$ such that, for each $b_V \in B_V$,...