# Questions tagged [linear-algebra]

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

4,210
questions

**-2**

votes

**0**answers

21 views

### finding area of intersection of a rectangle and circle [closed]

I have a rectangle intersecting with a circle. I want to find the area of intersection shown in blue in the figure.
I am calculating the area of rectangle as:
A = b x (50-d)
would that be the ...

**3**

votes

**1**answer

67 views

### Matrix inequality : trace of exponential of Hermitian matrix

I want to know whether the following inequality holds or not.
\begin{align}
(\mathrm{Tr}\exp[(A+B)/2])^2\leq(\mathrm{Tr}\exp A)(\mathrm{Tr}\exp B)\tag{1}
\end{align}
where $A, B$ are Hermitian ...

**0**

votes

**0**answers

46 views

### Image of Frobenius element under irreducible representation is diagonalizable

Let $K/ \mathbb Q$ be a Galois extension, and $\rho$ be an irreducible representation of the Galois group $Gal(K/ \mathbb Q)$. Consider an integer prime $p$ which doesn't ramify in $K$, and let $\...

**7**

votes

**2**answers

108 views

### Reference request: continuity of Cholesky factor

It most books dealing with Cholesky decomposition, or it is variants, one finds a statement of the form if $A$ is symmetric $k\times k$ positive semi-definite (non-negative definite) then the $k\times ...

**2**

votes

**1**answer

140 views

### The operator equation $AB = \lambda BA$ for self-adjoint operators

Suppose that $A$ and $B$ are self-adjoint bounded linear operators on a Hilbert space and $\lambda \in \mathbb{C}$. It turns out that if $\lambda \notin \{-1, 1\}$ then $AB=\lambda BA \implies AB = ...

**1**

vote

**0**answers

35 views

### Product of matrices has real eigenvalues?

Let $A$ be a (symmetric) positive definite matrix and $\hat{n}$ be an arbitrary unit vector. Consider $b,c,d,k$ arbitrary positive integers. I would like to know if the following matrix has real ...

**4**

votes

**0**answers

51 views

### Characterization of “PSD-Squared” Matrices

$\DeclareMathOperator\DNN{DNN}\DeclareMathOperator\CP{CP}$This question can be thought of as an offshoot of this MO question from a few months ago. Let $M_n(\mathbb{C})$ denote the set of $n \times n$ ...

**0**

votes

**0**answers

23 views

### Is the SVD optimal (by Eckart Young theorem) because maximal data variance is captured from both the column and row space?

I am asking this question seeking validation of an intuitive understanding of the veracity of the Eckart-Young theorem which struck me in my study of the SVD and Principle Component Analysis. The ...

**0**

votes

**0**answers

127 views

### Boundedness of total current in electrical network (Banded graph)

Consider the following symmetric matrix (adjacency matrix):
$$A=(a_{ij})_{1\leq i,j\leq n}$$
such that $a_{ij}=a_{ji}, a_{ii}=0$ and $a_{ij}=0$ for $|i-j|\geq k$ where $k\geq3$. We also have $1\leq a_{...

**0**

votes

**0**answers

43 views

### Relation between subspaces of diagonal matrix and its “sign” matrix

Let $D$ be a $n \times n$ diagonal matrix with both positive and negative (but all non-zero) entries. Let $J = sign(D)$ be the matrix of $1$s and $-1$s representing the signs of the entries of $D$. ...

**-1**

votes

**0**answers

21 views

### Express the length of a projection in terms of the dot product [closed]

I have this question when reading Linear Algebra http://www.math.nagoya-u.ac.jp/~richard/teaching/f2014/Lin_alg_Lang.pdf, page 39 - 40.
Consider a plane defined by the equation (X - P)· N == 0, and ...

**9**

votes

**2**answers

249 views

### Exponential decay of voltage potential difference

Consider the following adjacency matrix of a complete graph:
$$A=(e^{-|i-j|})_{1\leq i\neq j\leq n}$$
with 0 on the diagonal. Let $D=diag\{d_1,...,d_n\}$ be the degree matrix where $d_i=\sum_{j\neq i}...

**3**

votes

**1**answer

62 views

### Observable nearly commuting with a “complete” set of commuting observables

Consider the Hilbert space $H = E^{\otimes n}$ where $E=\mathbb{C}^2$.
On $E$ we have an observable $O$ (i.e. a Hermitian matrix) that is diagonalizable in the standard basis with eigenvalues $1$ and ...

**1**

vote

**0**answers

41 views

### On smooth extensions of functions

Let $f(x) = \left(I - \hat{n}\hat{n}^T \cdot\textbf{1}_{\vec{n}^TAx \geq 0}\right)Ax$, where $I$ is the identity matrix, $A$ is a (symmetric) $d\times d$ positive definite matrix, $\hat{n}$ is an ...

**5**

votes

**2**answers

197 views

### Rings $R$ such that every [regular] square matrix with entries in $R$ is equivalent to an upper triangular matrix

Let $\text{M}_n(R)$ be the ring of $n$-by-$n$ matrices with entries in a commutative unital ring $R$. Theorem III in
C.R. Yohe, Triangular and Diagonal Forms for Matrices over Commutative ...

**1**

vote

**0**answers

63 views

### Computational complexity in linear solvers

I have recently been trying out methods of coding for solving systems of linear equations on Python. Of course, I first used the inbuilt function $\mathit{inv}$ under certain if-conditions to obtain ...

**1**

vote

**1**answer

102 views

### Integrality certification for product of two matrices $A B^{-1}$

Let's consider two non-singular integer matrices $A,B \in\mathbb{Z}^{n\times n}$. I want a test to check if $A\times B^{-1}$ is integral (or no denominators). I am referring the unimodular ...

**-1**

votes

**0**answers

37 views

### “euler form” of a unitary matrix? [closed]

A little background:
I'm a student of physics (forgive me for a difference in language or the lack of familiarity with this topic), trying to solve an equation of the form:
\begin{equation}
\mathcal{...

**0**

votes

**0**answers

66 views

### Classical papers in linear algebra suitable for an undergraduate reading group?

I'm interested in collecting some of the classical papers in linear algebra over the years. To be more specific, I'm looking for interesting and useful results that extend beyond what is typically ...

**0**

votes

**0**answers

37 views

### trace expressions for matrix quadratic forms [migrated]

Let $A$ be a real symmetric $n \times n$ matrix. Which quadratic forms in $A$ can be written in trace form?
Such an expression would naturally generalise some invariant random matrix ensembles. ...

**-3**

votes

**0**answers

31 views

### Problems with rank of a matrix [closed]

I have some doubts about the rank of this matrix:$
\begin{bmatrix}\cos(x) & 0 & \frac{\sin(x)}{\cos(y)^2} \\
\sin(x) & 0 & \frac{\sin(x)}{\cos(y)^2} \\
\text{tg}(y) & \frac{1}{\cos(...

**1**

vote

**1**answer

94 views

### Symmetric tensor components

EDIT: I thought on rephrasing the question in another way:
I have been working recently with a tensor that satisfies
$A_{ijkl}=A_{i+b,j+b,k+b,l+b}$ $\forall$ $i,j,k,l$ $\in$ Z
$$dist(i,j,k,l)\...

**1**

vote

**0**answers

88 views

### Phase angles of a complex eigenvector

I have the following system for $\lambda \in \Bbb C, \lambda \neq 0$ and $\pmb{p},\pmb{q} \in \Bbb C^n$, $(\pmb{p}^T, \pmb{q}^T)\neq0$:
$$\begin{cases} F(\lambda) \pmb{p} - g(\lambda) \pmb{q} - \...

**2**

votes

**0**answers

93 views

+50

### Question on “semi-linear” dynamical systems

I am interested in understanding the convergence properties of a dynamical system at zero. We know that a dynamical system of the form $x_{k+1} = Ax_{k}$ where $A$ is a symmetric positive definite ...

**3**

votes

**1**answer

95 views

### Bounds on $\operatorname{sgn}(Au) - \operatorname{sgn}(Av)$ when $\|u-v\|_1 \leq \epsilon$

$\DeclareMathOperator\sgn{sgn}$Suppose A is a $N \times N$ Hermitian and unitary matrix, i.e., $A^{\dagger}=A$ and $A^{\dagger}A=I =AA^{\dagger}$. (Assume all entries are real.)
And let $u \in \{-1,1\...

**12**

votes

**0**answers

261 views

### Combinatorial proof of invertibility of a symmetric matrix associated to the ring of matrices over a finite field

Let $F$ be a finite field of $q$ elements with characteristic $p$. Let $M_n(F)$ be the ring of $n\times n$ matrices over $F$. We define a $q^{n^2}\times q^{n^2}$ symmetric matrix $L$ over the ...

**-1**

votes

**0**answers

81 views

### Question on eigenvectors [closed]

Let $\hat{n}$ be a unit vector and $\Delta$ a diagonal matrix of size $m\times m$ with positive entries (at least one entry is greater than one). I would like to show that the sequence of matrices $(I ...

**0**

votes

**0**answers

14 views

### Proof for strict separation of the eigenvalues of a Jacobian matrix with its minors

Let's consider a jacobi matrix (or tridiagonal symetric matrix where adjacent diagonals coefficients are strictly positive) :
\begin{equation}
T_n =
\begin{bmatrix}
a_1 & b_1 & 0 & \...

**1**

vote

**2**answers

229 views

### Linear independence of exponential functions: a reference

Is there a publication containing this obvious fact: For any real $T>0$, any natural $n$, any complex $c_1,\dots,c_n$, and any distinct complex $z_1,\dots,z_n$ such that $\sum_1^n c_k e^{tz_k}=0$ ...

**1**

vote

**0**answers

57 views

### Computational complexity of computing the trace of a matrix product under some structure

I have two problems related to computing some trace, and some (possibly suboptimal) answers. My question is about a potential more efficient algorithm for each one. (More interested in an answer to ...

**6**

votes

**2**answers

222 views

### Characteristic polynomial of checker matrix

For every integer $n > 0$, let $C_n$ be the $4n \times 4n$ matrix having $1$'s in all positions $(i, j)$ such that $i - j$ is even, $3$'s in the two diagonals determined by $|i - j| = 2n + 1$, and $...

**0**

votes

**4**answers

223 views

### Solution for $Xa + X^Tb = c$ where $X^TX = I$?

There are three known $n\times1$ vectors: $a, b, c$, along with one unknown $n\times n$ matrix: $X$. I am only interested in the $n={2,3}$ cases.
$X$ is $2\times 2$ or $3\times 3$ rotation matrix ...

**6**

votes

**1**answer

79 views

### Is the matrix positive definite given the Gauss-Seidel method converges?

I know that the Gauss-Seidel method converges given that the matrix you want to solve is symmetric positive definite. However, I'm wondering if the "converse" of the statement is true. Namely, if $A$ ...

**4**

votes

**0**answers

214 views

### Inequalities for trace/eigenvalues of product of multiple 2x2 matrices

Consider the matrix product $\prod_i^n A_i$,
where each $A_i$ is a $2\times2$ matrix having the form $A_i = \left( \begin{smallmatrix} \lambda + \alpha_i & -\beta_i \\ 1 & 0\end{smallmatrix}\...

**4**

votes

**1**answer

154 views

### Neighborhood of an orthogonal matrix

Let $A\in O(n)$ be an orthogonal matrix and let $\vec{a}_1,\dots,\vec{a}_n$ be its rows. For a vector $\vec{v}=[v_1,\dots,v_n]$, let $\max(\vec{v})=\max\{|v_1|,\dots,|v_n|\}$. Prove or disprove that ...

**1**

vote

**0**answers

41 views

### A real system of bilinear equations with $2n$ unknown and equations

I have the following system of $2n$ bilinear equations, for a square invertible matrix $A \in \mathbb{R}_{n \times n}$, and $2n$ unknowns organized in vectors $x,y \in \mathbb{R}^n$:
$$
diag(y) A x = ...

**0**

votes

**0**answers

32 views

### pseudo-inverse of short fat matrix

Consider the matrix $A \in \mathbb{R}^{m \times n}$ with $m < n$.
Is its pseudo-inverse $A^\dagger = (A^\top A)^{-1} A^\top$ computable ?
I'd expect not, because $A^\top A$ is not of full rank ( $...

**0**

votes

**1**answer

57 views

### Similar to inverse plus rank 1

Given a real, invertible matrix $A$. For which vectors $b$ and $c$ is
$$
A^{-1} + bc^T
$$
similar to $A$? And is the rank-1 matrix $bc^T$ unique?

**4**

votes

**1**answer

94 views

### Rank of a block of an invertible matrix

Let $A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}$ be an invertible matrix where $A_{11}$ is square. Let $A^{-1} =: B = \begin{bmatrix} B_{11} & B_{12} \\ B_{21} &...

**-2**

votes

**0**answers

57 views

### Can we use modified Cholesky's decomposition for solving linear equations?

So I was learning about modified Cholesky's decomposition, specifically Gill, Murray, and Wright algorithm. And although I was able to decompose a negative definite matrix into $LDL^T$, when I used ...

**3**

votes

**1**answer

146 views

### An orthogonal matrix that satisfies a property must be a permutation matrix

Let $A$ be an $n\times n$ orthogonal matrix such that $\sum_{k=1}^na_{ik}^3a_{jk}=\sum_{k=1}^na_{jk}^3a_{ik}$ for every $1\le i,j\le n$.
Original question which is solved by a counterexample given (...

**1**

vote

**1**answer

42 views

### How can I find minimum and maximum eigenvalue of non-positive define matrix [closed]

There is a power iteration method, but it only returns the greatest(in absolute value) eigenvalue of matrix. So when we have negative eigenvalues it'll give wrong results.
Is there any method, which ...

**4**

votes

**0**answers

109 views

### Number of elements in $\mathrm{GL}(n,p)$ with maximal order

I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$.
I would like to know how many matrices have an order of exactly $p^n -1$. Do they ...

**0**

votes

**1**answer

37 views

### Lipschitz continuity of multivariable function in expected value

Suppose $h:\mathcal{X \times Y \times W}\rightarrow\mathcal{X}$, with $\mathcal{X,Y,W} \subseteq \mathbb R^d$, $d \in \mathbb N$ is Lipschitz in $x,y$, i.e.,
$$ \| h(x,y,w) - h(x',y',w) \|_2 \le L_h (...

**1**

vote

**0**answers

41 views

### Diagonally similar to submatrix of orthogonal matrix

Given $A \in \mathbb{R}^{n \times n}$, with $0 < |\det(A)| < 1$. Does a diagonal matrix $D$ exist such that
$$
B = D^{-1} A D
$$
is the principal submatrix of an $(n+1)\times(n+1)$ orthogonal ...

**0**

votes

**0**answers

66 views

### How to solve a non-local self-consistent equation

I have been struggling lately with solving numerically an equation of the form:
$$ g(x\pm x_{0}) = F[ g(x) ] $$
where $g(x)$ is a matrix satisfying the condition $g(x\to\pm\infty)=0$. My question is ...

**1**

vote

**0**answers

35 views

### Fastest way to calculate the eigenvalues of a product of two Toeplitz matrices

I have the following problem:
I need to find the fastest way to calculate the eigenvalues of a matrix that is the product of two Toeplitz matrices. $B = A U$.
The first is a regular Toeplitz matrix $A$...

**2**

votes

**1**answer

142 views

### How to find eigenvalues following Axler?

Preparing my Linear Algebra lecture I like the determinant free approach of Axler because the proof that operators $T$ on an $n$-dimensional complex vector space have eigenvalues is so simple:
Fix ...

**1**

vote

**1**answer

62 views

### $\arg\max$ in the dual norm of the nuclear norm

Given a matrix $X \in \mathbb{R}^{m \times n},$ then the spectral norm is defined by
$$\left \| X\right\| := \max\limits_{i \in \{1, \dots, \min\{m,n\}\} }\sigma_i (X)$$
whereas the nuclear norm is ...

**0**

votes

**0**answers

60 views

### Probability that random points are affine subspace

I asked this question in Math stack exchange, but I think it is more relevant here.
Suppose that $\mathbb{F}_q$ is a finite field with $q$ elements. Let $U = \{u_1, \ldots,u_m\}$ be a set of $m$ ...