# Questions tagged [linear-algebra]

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

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### Maximum number of vectors with bounds on inner products

Suppose there are 2n vectors $\{m_1,m_2,...,m_n\}$ and $\{\mu_1,\mu_2,...,\mu_n\}$. All vectors are in k-dimensional Euclid space $R^k$. The vectors satisfy: $$m_i \mu_j \leq 0, \quad \forall i<j$$...
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### Inverse of a matrix defined as a quadratic form

Suppose I have a vector $x$ of $n$ reals, and tensors $A,B$ of respective sizes $(n,k,k,n)$ and $(k,k,n)$, so that both the array-vector product $Bx$ and the quadratic form $x'Ax$ are of size $(k,k)$. ...
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### Deviation of random matrix from its expectation informs the positiveness of its second smallest eigenvalue

Let $A$ be a PSD random matrix, which has $0$ as one of its eigenvalues. The second smallest eigenvalue of the expectation of $A$ writes as $\lambda_2(\mathbb{E}(A))>0$. Why the following statement ...
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### Testing a condition in linear algebra involving Krylov subspaces

Let $A \in \mathcal{M}_n(\mathbb{R})$ be a real-valued $n \times n$ matrix. For $b \in \mathbb{R}^n$, I consider the Krylov subspace $$K_A(b) = \operatorname{span} \{ b, A b, \dotsc, A^{n-1} b \}.$$ ...
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### Reference for (general case) of uniqueness of singular value decomposition (SVD)

My statistics research requires me to understand the non-uniqueness of SVD in the degenerate case of repeated singular values. I believe that the statements and proofs on this StackExchange posts are ...
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### Action of complex torus on a vector space

Consider a torus $T$ over $\mathbb{C}$. Let $\rho: T\rightarrow \operatorname{GL}_{n}(\mathbb C)$ be a finite dimensional complex representation. Is there an elementary way (undergrad level) to see ...
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### The eigenvalues of the matrix $\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$

What are the eigenvalues/eigenvectors of the matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$ when $n$ is odd?
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### Can the Jordan decomposition of a matrix be computed in a backwards stable way?

Let $PJP^{-1}$ denote the Jordan decomposition of $M$. The matrix $J$ is a direct sum of Jordan blocks; it is unique up to permutation of the Jordan blocks. The matrix $P$ is not unique. There are two ...
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### The dimension of the eigenvector space of non-negative irreducible matrices

Let $A$ be an irreducible non-negative matrix. Is it true that the eigenvectors of $A$ can span the $R^n$ ? Or are all the eigenvalues of $A$ distinct?
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### A question on linear algebra over non-Archimedean local field

Let $\mathbb{F}$ be a non-Archimedean local field. Let $\{T_a\}_{a=1}^\infty$ be a sequence of linear operators $\mathbb{F}^n\to\mathbb{F}^n$ of rank $n$. After a choice of subsequence, is it ...
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### Is there a redundant constraint in linear programming? [closed]

From wikipedia: But... Why do we need the $x\ge 0$ part? We can instead do $-x\le 0$, and thus saving a line in the definition (which is not a big deal but nevertheless nice). (In order to do that, ...
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### What information concerning the eigen-structure are transformed on the antidiagonal submatrices?

Let us fix symmetric matrices $A_1$ $A_2$ in $M_m(\mathbb{R})$ with $A^2_1=\alpha I$ and $A^2_2=\beta I$ for some positive $\alpha ,\beta$. For a given matrix $B\in M_m(\mathbb{R})$, let us ...
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### Positive linear recurrent sequence

Suppose there is a linearly recurrent sequence $a_k$ satisfies $a_k\geq 0$ and $\sum_{k=1}^{\infty}a_k=1$. Can we always find a $x$ and $r<1$ such that $a_k\leq x r^k$?
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### How to decompose a matrix over a ring $F[X_1,\ldots,X_k]$ as a product of two matrices

Let $F$ be a field. Assume any reasonable conditions if needed, such as $F=\mathbb R$, $F=\mathbb C$, $F$ is a finite field, or $F$ has a specific characteristic, etc. Let $C$ be an $n\times1$ matrix ...
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### Tensor nuclear norm for a binary 3rd-order tensor

I am interested in the low-rank approximation of a binary(01) third-order tensor. Does anyone know how to define its tensor nuclear norm based on whatever tensor decomposition methods? Could anyone ...
### Algorithm for generalized Hilbert's Theorem 90 over $\Bbb R$
$\newcommand{\GL}{\operatorname{GL}} \newcommand{\R}{{\Bbb R}} \newcommand{\C}{{\Bbb C}}$For a natural number $n$, let $z\in \GL(n,\C)$ be a 1-cocycle of $G=\GL_{n,\R}\,$, that is, an invertible ...
Given the function $$E(M) = \sum_{i=1}^N \sum_{a=1}^K \left( M_{ia} \cdot \left\lVert\sum_{i=1}^N M_{ia}\cdot x_i\right\rVert_2^2 \right)$$ $x$ is a given constant matrix, $x_i$ is a the \$n_\text{th}...