# Questions tagged [linear-algebra]

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

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### 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 ...
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### 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 ...
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### 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 $D$. ...
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### 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 ...
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### 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 ...
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### 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 & \...
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### 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 $\sum_1^n c_k e^{tz_k}=0$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $A$...
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### 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 ...
### $\arg\max$ 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 ...
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 $m$ ...