# Questions tagged [linear-algebra]

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

3,859
questions

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23 views

### 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{...

**1**

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14 views

### 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\...

**15**

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248 views

### 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)...

**4**

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49 views

### 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 $2\times 2$ ...

**5**

votes

**1**answer

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 ...

**2**

votes

**0**answers

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 ...

**1**

vote

**1**answer

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 ...

**0**

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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, $\|v\|_2=1$....

**1**

vote

**1**answer

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 ...

**2**

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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 ...

**2**

votes

**1**answer

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|^\...

**-3**

votes

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69 views

### 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 ...

**1**

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252 views

+50

### 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 ($\...

**-1**

votes

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21 views

### 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 $B=\{u_1,...,u_n\}$...

**2**

votes

**1**answer

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,
$...