# Questions tagged [linear-algebra]

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

**-3**

votes

**0**answers

16 views

### Unable to understand that why dot product of columns is not zero for vectors (2,-2,1) and (1,2,2,) [on hold]

I am taking a linear algebra course on Udemy and in topics of Orthogonal projection a topic of zero dot products comes up where instructor explains that dot products of columns (2,-2,1) and (1,2,2,) ...

**0**

votes

**0**answers

11 views

### A right-inverse property of a nonlinear optimization problem

Disclaimer: This might be a silly question. However, after some days of thought, I could not find a clear/rigorous answer. So I decided to post it here.
Let $Y\in\mathbb{R}^{n\times p}$ and $X\in\...

**0**

votes

**0**answers

16 views

### An two-norm estimate for symmetric $k$-tensors

Let $(V, \langle, \rangle)$ be an $n$ dimensional innerproduct space and let $S^k(V)$ denote the space of $k$-fold symmetric tensors. The inner product naturally extends to $S^k(V)$. Denote the ...

**0**

votes

**0**answers

17 views

### Stability analysis with minimal spectral norm

Let $A \in \mathbb{R}^{n \times n}$ with
$$
s(A) = \inf_{D \textrm{ is diagonal}} \| D^{-1} A D \|_2 > 1
$$
Does there exists $m \in \mathbb{N}^n$ and $z \in \mathbb{C}$ with $|z| > 1$ such ...

**1**

vote

**0**answers

16 views

### How explicitly write a projective transformation between the conics over the univariate function field?

Consider the quadratic forms
$$
Q_1 = x^2 + y^2 - (t^2+1)z^2,\qquad Q_2 = x^2 + y^2 - z^2
$$
over the rational function field $\mathbb{F}_p(t)$, where $p > 2$ is a prime such that $t^2 + 1$ is ...

**-1**

votes

**0**answers

19 views

### relevant probability for vector in multi-dimensional spaces [on hold]

is it possible to compare the probability that for example a 2D vector lies on top of an other 2D vector within the 2D space, with the event that a 3D vector lies inside a 2D plane of the same 3D ...

**0**

votes

**1**answer

35 views

### Intersection between a line and a n dimensional parallelotope

Suppose that I have a line in an $n$ dimensional space described by
$$ X=A+Bk, \quad \quad X,A,B \in \mathbb{R}^n, k \in \mathbb{R} $$
here $A$ is known and I want to find all the possible vectors $B$ ...

**3**

votes

**0**answers

35 views

### Minimize Frobenius instead of Spectral norm via diagonal similarity

Given square matrix $A$. I am looking for a numerical solution for
$$
s(A) = \inf_D \| D^{-1} A D\|_2,
$$
where $D$ is a non-singular, diagonal real matrix. A numerical solution was here. However, ...

**1**

vote

**1**answer

63 views

### Inequality for the operator norm of a product of matrices

I am working with a product of $n\times n$ matrices $A_1,\ldots,A_k$. Under which conditions can I assume that
$$\|A_1\cdots A_k\|_\infty \leq \|A_1\cdots \hat{A_i}\cdots A_k\|_\infty \|A_i\|_\infty,...

**2**

votes

**1**answer

100 views

### counting invertible matrices [on hold]

Let $T$ be a subset of vector space $Z_2^n$ and $A$ be an element of $GL(2,n)$ means invertible matrices with entries $\{0,1\}.$ Let $T$ be invariant under A. It means for any $t \in T$, $tA \in T$. ...

**4**

votes

**2**answers

219 views

### The Aleksandrov-Fenchel inequality of mixed discriminants for Hermitian matrices

Suppose $A,A_1,\ldots,A_{n-2}$ (resp. $B$) are (resp. is) real positive-definite (resp. arbitrary) symmetric $n\times n$ matrices and denote by $D(\cdot,\ldots,\cdot)$ the mixed discriminant. We have ...

**7**

votes

**0**answers

479 views

### Can we write unitary matrices as positive linear combinations of Hermitian matrices?

The space $M_n:=M_n(\mathbb{C})$ of complex $n\times n$ matrices has the structure of a finite-dimensional complex vector space.
The space of Hermitian matrices forms a cone in this vector space $M_n$...

**5**

votes

**0**answers

87 views

### Matrix operations preserving the middle coefficients of characteristic polynomial

Let $n$ be a positive integer. For a ring $A$ and matrix $M \in \mathrm{Mat}_{n \times n}(A)$, let $\chi_{M}(t) = \det(M-t \operatorname{id}_{n}) = (-1)^{n}(t^{n} - \sigma_{M,1}t^{n-1} + \dotsb + (-1)^...

**7**

votes

**2**answers

781 views

### An Expectation of Cohen-Lenstra Measure

The Cohen-Lenstra measure on the set of abelian p-groups assigns $\mathbb{P}(G) = \prod_{i \geq 1} \left( 1 - \frac{1}{p^i}\right) \cdot |\mathrm{Aut}(G)|^{-1} $. Apparently, this is equivalent to ...

**0**

votes

**0**answers

50 views

### Quadrics over the univariate function field with discriminant of minimal degree

Consider a non-degenerate quadric $Q(x,y,z) \subset \mathrm{P}^2$ over the univariate function field $\mathbb{F}_p(t)$, where $\mathbb{F}_p$ is a prime finite field, $p > 2$. For simplicity assume ...

**5**

votes

**1**answer

288 views

### A surprising identity: $\det[\cos\pi\frac{jk}n]_{1\le j,k\le n}=(-1)^{\lfloor\frac{n+1}2\rfloor}(n/2)^{(n-1)/2}$

On the basis of my computation, here I pose my following conjecture involving the cosine function.
Conjecture. For any positive integer $n$, we have the identity
$$\frac1{2n}\det\left[\cos\pi\frac{jk}...

**6**

votes

**4**answers

3k views

### Maximum determinant of $\{0,1\}$-valued $n\times n$-matrices

What's the maximum determinant of $\{0,1\}$ matrices in $M(n,\mathbb{R})$?
If there's no exact formula what are the nearest upper and lower bounds do you know?

**0**

votes

**0**answers

24 views

### Intersection of parameterized cosets in a free-abelian group

Let $L_1,L_2$ be subgroups of $\mathbb{Z}^m$, and let $\mathbf{P}_1,\mathbf{P}_2$ be $r\times m$ integer matrices. Then, it is straightforward to check that the set
\begin{equation}
S_{\{1,2\}} := \{...

**26**

votes

**1**answer

505 views

### Determinants of binary matrices

I was surprised to find that most $3\times 3$ matrices with entries in $\{0,1\}$ have determinant $0$ or $\pm 1$. There are only six out of 512 matrices with a different determinant (three with $2$ ...

**3**

votes

**0**answers

183 views

### Invertibility of the Schur Complement

Suppose that
$$
M = \begin{bmatrix}A & B \\ C & D\end{bmatrix}.
$$
I know that if $D$ and $M\setminus D$ (where $M\setminus D$ is the Schur Complement of $D$ in $M$) are invertible, then $M$...

**2**

votes

**1**answer

79 views

### volume of parallelotope in $L^2(\mathbb R).$ [closed]

Let $L^2(\mathbb R)$ is complex Hilbert space with standard inner product.
Does it make sense to talk of volume of parallelotope formed by following vectors in $L^2(\mathbb R):$ say, e.g.,
$$\{ f(...

**7**

votes

**1**answer

257 views

### Coordinate free isomorphism between $d+1$-dimensional antisymmetric rank $2$ tensors and $d$-dimensional symmetric rank $2$ tensors

The space of $(d+1)$-dimensional antisymmetric matrices has the same dimension as the space of $d$-dimensional symmetric matrices, $\frac12d(d+1)$. There are isomorphisms between the two spaces, e.g. ...

**8**

votes

**0**answers

363 views

### Prove the optimality of the following constant

Let $E$ be a complex Hilbert space.
In (arXiv) it was shown that for $A=(A_1,...,A_n) \in \mathcal{B}(E)^n$ we have,
$$\displaystyle\frac{1}{2\sqrt{n}}\|A\|\leq \omega(A) \leq \|A\|,$$
where
$$
\...

**0**

votes

**0**answers

27 views

### Invertible matrices T that commutate with the Jordan matrices A of size n [closed]

I'm supposed to find the set of invertible matrices T that commutate with the Jordan matrices A of size n.
Through trial and error, I've found that tᵢ₊₁,ₖ = tᵢ,ₖ₋₁. I haven't found a way to prove ...

**2**

votes

**1**answer

172 views

### Are Linear Maps resistant to Noise?

Let's assume I have a $m \times m$ matrix $M$ with Frobenius norm $1$ and a unit vector $x \in S^{m-1}$. I also have a second $m \times m$ matrix $M^*$ which is obtained from the first one plus some ...

**3**

votes

**0**answers

124 views

### Combinatorics question

Let $A = (a_{ij})_{1\le i,j\le h}$ be an $h$-by-$h$ non-degenerate upper triangular matrix with entry $a_{11} = 1$. Let $\Phi = \{\alpha_1,\alpha_2,\ldots,\alpha_d\}\subseteq \{1,2,\ldots,h\}=I$ be an ...

**18**

votes

**4**answers

2k views

### Jacobi's equality between complementary minors of inverse matrices

What's a quick way to prove the following fact about minors of an invertible matrix $A$ and its inverse?
Let $A[I,J]$ denote the submatrix of an $n \times n$ matrix $A$ obtained by keeping only the ...

**1**

vote

**1**answer

139 views

### On $\det[x+(\frac{i\pm j}p)]_{1\le i,j\le(p-1)/2}$ for primes $p\equiv 3\pmod 4$

I have made the followng conjecture on the basis of my computation.
Conjecture. For any prime $p\equiv3\pmod4$ with $p>3$, we have
$$\det\left[x+\left(\frac{i+j}p\right)\right]_{1\le i,j\le(p-1)/2}...

**0**

votes

**1**answer

57 views

### Maximum of a sum of Gaussian functions

Consider the function which maps $\mathbb{R}^n$ to $\mathbb{R}$
\begin{align}
f(x) = \sum_{i=1}^{n} b_i\phi_i(x)
\end{align}
where $\phi_i(x) = \exp(-\frac{||x-x_i||_2^2}{2})$ are Gaussian functions ...

**5**

votes

**1**answer

278 views

### $(AB)^+\approx B^+A^+$ for $B$ “fat” enough?

Let $A\in\mathbb{R}^{r\times m}$ be a matrix of full row rank, and let $\cdot^+$ denote the Moore-Penrose inverse.
Consider a sequence of matrices $\{B_n\}_{n>1}$, $B_n\in\mathbb{R}^{m\times n}$, ...

**3**

votes

**0**answers

67 views

### How to find the best similarity transformation between two symmetric matrices $\mathbf{A}$ and $\mathbf{B}$? [duplicate]

Suppose I have two matrices $\mathbf{A}\in\mathbb{R}^{n\times n}$ and $\mathbf{B}\in\mathbb{R}^{n\times n}$. I want to know what's the best similarity transformation between these two matrices when we ...

**7**

votes

**2**answers

249 views

### Direct product of free groups in $\mathrm{SL}_3(\mathbb{Z})$

Let $\mathbb{F}_2$ be the free group on two generators. Does $\mathbb{F}_2 \times \mathbb{F}_2$ embed as a subgroup of $\mathrm{SL}_3(\mathbb{Z})$?

**3**

votes

**1**answer

119 views

### Endomorphisms of the p-adic group $(\mathbb Z_p,+)$

Does there exist an endomorphism of $(\mathbb Z_p,+)$ of finite order different of $x\mapsto\xi x$ where $\xi$ is a root of unity in $\mathbb Z_p$?
Thanks in advance

**2**

votes

**2**answers

51 views

### Lower bound of positive entropies of automorphisms on tori

Let $A$ be an automorphism on tori $\mathbb{T}^d$. It is well known that the topological entropy
$$
h(A)=\sum_{\lambda} \max\{0, \log|\lambda| \}
$$
where $\lambda$ goes through all eigenvalue of $A$ ...

**3**

votes

**1**answer

150 views

### Volume of polyhedron

Given the following polyhedron: All the $n\times n$ matrices $\boldsymbol{X}$ with elements $x_{ij}\in(0,1)$ such that
$$\boldsymbol{X}\cdot\boldsymbol{1}=\boldsymbol{r}, \boldsymbol{1}^T\boldsymbol{...

**5**

votes

**4**answers

385 views

### Why do some linear cellular automata over $Z_{2}$ on the torus have small order?

At https://dmishin.github.io/js-revca/index.html, you can play around with reversible cellular automata. I noticed that on that site, that for the reversible linear cellular automata (which I have ...

**0**

votes

**0**answers

62 views

### Representation of symmetric group as Cremona transformations

Question from me and a colleague:
Given a matrix
\begin{equation}
U =
\begin{bmatrix}
U_{11} & U_{12} \\
U_{21} & U_{22}
\end{bmatrix}
\quad \text{with } U_{22} \neq 0,
\end{equation}
...

**0**

votes

**1**answer

26 views

### tensor stability of block-positive matrices

Let $X_{AB}$ be an operator acting on the tensor-product Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$. Suppose that $X_{AB}$ is block positive, meaning that (in Dirac notation)
$\langle \psi |...

**2**

votes

**1**answer

47 views

### Lie-algebra-like relation for totally symmetric 4-tensors

There are many totally symmetric real 4-tensors, $T_{ijkl}$, which satisfy the relation
$$T_{ijmn}T_{mnkl} + T_{ikmn}T_{mnjl} + T_{ilmn}T_{mnjk} = c T_{ijkl}$$
with some constant $c$. By the way ...

**4**

votes

**0**answers

113 views

### How to formulate supercommutativity in a characteristic free way?

I would never dare posting this here, but the question https://math.stackexchange.com/q/3019853/214353 on math.SE did not receive any feedback (except for 13 views and one upvote) since November 30, ...

**6**

votes

**0**answers

66 views

### mean distance between subspaces

Consider the Haar measure $\mu$ on the Grassmannian $G(n, k)$ of $k$-dimensional subspaces in $\mathbb{R}^n.$ Now, pick pairs of subspaces uniformly at random with respect to $\mu,$ and compute their ...

**1**

vote

**0**answers

67 views

### Shortest vectors in tensor product and maximal lattices in tensor product

$\mathcal L$ and $\mathcal L'$ be full rank lattices in $\mathbb R^n$ with shortest vectors $v_1,\dots,v_n$ and $v_1',\dots,v_n'$ respectively where $$\|v_1\|_2\leq\dots\leq\|v_n\|_2$$
$$\|v_1'\|_2\...

**24**

votes

**5**answers

2k views

### Computing a determinant involving roots of unity

Let $d \geq 2$ be an integer and $\xi=\exp(\frac{2\pi i}{d})$. I am trying to compute the determinant of the matrix
$$
(\xi^{ij}-1)_{1 \leq i, j \leq d-1}.
$$ Let me call it $\Delta(d)$. For small ...

**4**

votes

**2**answers

144 views

### Hyperrectangle that contains most of cube's interior (except its vertices)

Let $n>0$, and let $p,q\in (0,1)$ such that $p<q$.
Is there a hyperrectangle $H$ that satisfies the following:
$\forall i\in{1,\dots,n}:\\ H\supset \prod_{j=1,\dots,n}
\begin{cases}
[p,q], &...

**0**

votes

**1**answer

104 views

### Smallest collection of linear operators satisfying isometry property

Let $\mathscr{A}=\{(A_{1,1},A_{1,2},A_{1,3}),...,(A_{S,1},A_{S,2},A_{S,3})\}$ be a collection of linear operators $A_{n,k}:\mathbb{R}^2 \rightarrow \mathbb{R}^2$. For $u$ and $v\in (\mathbb{R}^2)^3$, ...

**1**

vote

**2**answers

155 views

### How to compute inverse of sum of a unitary matrix and a full rank diagonal matrix?

$C = A+D$, $A$ being a unitary matrix and $D$ a full rank diagonal matrix. Is there any easy way to compute $C^{-1}$ from $A^{-1}$ and $D$, if it exists?
I am interested in this question, because my ...

**-4**

votes

**0**answers

145 views

### On the determinant $\det[(i^2+dj^2)(\frac{i^2+dj^2}p)]_{1\le i,j\le(p-1)/2}$ with $p$ an odd prime

Let $p$ be an odd prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. We define the determinant $D(d,p)$ by
$$D(d,p):=\det\left[(i^2+dj^2)\left(\frac{i^2+dj^2}p\right)\right]_{1\le i,j\le(p-1)/2}...

**2**

votes

**1**answer

144 views

### The effect of random projections on matrices

Let $A\in\mathbb{R}^{n\times n}$ be a given normal matrix, i.e. $A^TA=AA^T$. Let $P_s\in\mathbb{R}^n$ be a random projection matrix to an $s$-dimensional subspace in $\mathbb{R}^n$.
Suppose $\frac{A+...

**237**

votes

**25**answers

41k views

### Geometric Interpretation of Trace

This afternoon I was speaking with some graduate students in the department and we came to the following quandry;
Is there a geometric interpretation of the trace of a matrix?
This question should ...

**1**

vote

**1**answer

431 views

### Conjecture that relates matrix systems with some polynomials of integer coefficients as solution sets

Assume $x$ is a variable belongs to $\mathbb R \setminus \{ 0,-1,+1 \}$ and consider for all $i, j \in \mathbb N$,
$$a(i,j) = \frac{(x^{i+1} + 1)^{j-1} + (x-1)}{x}$$
then for all $n \in \mathbb N$ ...