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      Questions tagged [linear-algebra]

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

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      Converting a vector in a cone statement to inequality constraints

      I would like to convert the following condition for $x$ \begin{align} x = N \lambda, \lambda \geq 0 \end{align} to a pure linear inequality form, i.e. find an $L$ and eliminate $\lambda$ \begin{...
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      Weird subspace/equality-constrained LP problem/variant of change-making problem

      Assume that we have a set, $\mathscr{R}$ containing $m$-dimensional vectors. Solve $$\sum_{i=1}^n c_i\leq\delta$$ $$\text{subject to } \sum_{i=1}^n r_i c_i=x^\prime \text{ for all }x^\prime$$ where $0\...
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      A curious switch in infinite dimensions

      Let $V$ be a finite dimensional real vector space. Let $GL(V)$ be the set of invertible linear transformations, and $\Phi(V)$ be the set of all linear transformations. We can also characterize $\Phi(V)...
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      Low-rank approximation over finite fields

      Consider the finite field $\mathbb{F}_p$ with prime $p$. Let $I=\{ 0,1,2,...,|I|-1 \}\subset \mathbb{F}_p$ be an "interval". What can be the largest size $|I|$, such that there exists a $2\times 2$ ...
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      186 views

      Inequality involving tensor product of orthonormal unit vectors

      Let $e_1,...,e_r$ be the first $r$ standard basis of $\mathbb{R}^n, r<n$. Let $u_1,...,u_n$ be another orthonormal basis of $\mathbb{R}^n$. Let $\otimes$ be the tensor product on $\mathbb{R}^n$ and ...
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      121 views

      Is there literature on the study of “eigenmatrices”?

      Starting with a disclaimer: I will not be able to describe this with the correct terminology because I am trying to find the literature which I am unsure of it's existence. I will try to explain what ...
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      59 views

      Finding the closest special orthogonal matrix in Frobenius norm sense

      Given a $3\times3$ matrix $M$, if we would like to get the closest $\mathrm{SO}(3)$ matrix $R$ that minimizes \begin{equation} \|R-M\|_F \end{equation} then $R$ = $UV^{T}$ where $U$ and $V^{T}$ are ...
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      39 views

      Minimize the quadra inverse for of PD matrices

      I want to minimize the relation, $v^T (A+I+UQU^*)^{-1} v$, subject to $Q$ and $A$ are positive semi-definite, ${\rm trace}(Q)<1$. Here, $v$ is a random vector with unit norm, that is, $\|v\|_2=1$....
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      42 views

      Rank and edges in a combinatorial graph?

      Fix a $d\in\mathbb N$ and consider the matrix $M\in\{0,1\}^{2^d\times d}$ of all $0/1$ vectors of length $d$. Consider the matrix $G\in\{0,1\}^{n\times n}$ whose $ij$ the entry is $0$ if inner product ...
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      49 views

      Iterative algorithms for computing the kernel of a matrix

      Suppose $A$ is an $m \times n$ matrix in the form $$A=\begin{pmatrix} — a_1 —\\ — a_2 —\\ \vdots \\?— a_m — \end{pmatrix}$$ where $a_i \in R^n$ is the $i$-th row of $A$. It is possible to determine ...
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      1answer
      86 views

      Intersection of a lower dimensional space and a discrete set

      Let $H\subset \mathbb{R}^n$ with dimension ${\rm dim}(H)=\ell<n$; let $S$ be a finite subset of reals. My question is the following. Is it correct to say, $$ {\rm card}(H \cap V)\leqslant |S|^\...
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      The reign of matrices [closed]

      Suppose square matrices of order 3 are constructed whose elements are taken from the set A={1,2,3,4,5,6,7,8,9} without repetition and let M be the square matrix having maximum value of the ...
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      Is this function quasi-concave?

      Let $$ \mathbf{u} := \left( \mathbf{X}^H \mathbf{X} + \mathbf{I}_m + \mathbf{\lambda}\mathbf{D} \right)^{-1} \mathbf{X}^H \mathbf{y} $$ where $\mathbf{X}$ is $n \times m$ semi-orthogonal matrix ($\...
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      Shared basis of eigenvalues for two unitary transformations [migrated]

      I have two unitary transformations: $T$ and $S$ in a unitary space, and I know $TS=ST$. I need to prove that $T$ and $S$ have a shared basis composed of eigenvectors of both. i.e if $B=\{u_1,...,u_n\}$...
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      104 views

      Expansions of iterated, or nested, derivatives, or vectors--conjectured matrix computation

      The entry OEIS A139605 (also related OEIS A145271) has a matrix computation for the partition polynomials that represent the expansions of iterated derivatives, or vectors in differential geometry, $...

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