Questions tagged [linear-algebra]
For questions on linear algebra, including vector spaces, linear transformations, systems of linear equations, spanning sets, bases, dimensions and vector subspaces.
130,237 questions
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Relationship between the last columns of V1 and V2 in the Smith form decomposition of an integer matrix
Let $A$ be an invertible matrix with integer coefficients. Suppose we have two decompositions of $A$ into its Smith form $S$:
$U_1 A V_1=U_2 A V_2=S=\text{diag}\{1,\ldots,1,m\}$
where $S$ is the Smith ...
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Methods to find the minimum volume ellipse containing a convex set in $\mathbb{R}^2$
Consider a closed convex hull of a finite list of vertices $H \subset \mathbb{R}^2$ with a point $c = (x,y) \in H$. Given are a matrix $X \in \text{GL}(2, \mathbb{R})$ with columns that indicate the ...
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Is this approach to proving the multilinearity of the determinant valid (using Smith normal form)?
I'm working on an exercise in linear algebra involving the multilinearity of the determinant. The task is:
Let $𝐾$ be a field. Consider the determinant as a function:
\begin{align}
&\det : K^n \...
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Determinants are multilinear [closed]
I have an idea here to show, for a scalar $a$ and a vector $v$, that $f(a \cdot v)=a \cdot f(v)$. It is with the Smith Normal form and we can actually have $\det (s_1|\ldots|S_{j-1}| X |S_{j+1}|\ldots|...
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A strange question on sum of intersections of Lagrangian subspaces equal to intersection of sum of them
In research I need to prove a very strange question about the sum of intersections of Lagrangian subspaces being equal to intersection of sum of Lagrangian subspaces. Why I say strange is that it ...
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Putnam 2024 — Determinant of a Hankel matrix
Problem A6 on the 2024 William Lowell Putnam Mathematical Competition was as follows.
Let $c_0, c_1, c_2, \dots$ be the sequence defined so that $$ \frac{1 - 3x - \sqrt{1 - 14x + 9x^2}}{4} = \sum_{k=...
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Does ${\rm tr} (A^n)=1$ for all positive Integers $n$ imply $A$ has a single non-zero eigenvalue of 1? [closed]
$A$ is a Hermitian matrix on $\mathbb{C}^d$. If ${\rm tr}(A^n)=1$ for any positive integer $n$, can I say that $A$ has exactly one non-zero eigenvalue, which is 1?
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Interpretation of matrix product $AB$ as sum of matrices of rank $1$
Singular value decomposition of a matrix $A$ of dimension $m \times n$ reads $A=U\Sigma V^+$. It can be seen as a way to decompose the matrix into a sum of rank-1 matrices:
$$A=\sum_{i=1}^r \lambda_i ...
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How to rotate an inequality about a point in desmos [closed]
I am a 7th grader that is trying to finish a graphing project. I was planning on making the iconic F-14 and was trying to graph it in desmos. However, I am using inequalities to shade the wings. I was ...
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Is there a name for this kind of generalized eigenvalue problem
I've been thinking about a particular kind of "generalized eigenvalue", in the following sense:
Suppose we have two vector spaces $V_1$ and $V_2$, and let $A$ be a matrix over $V_1$. The &...
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Probability that a linear combination of given matrices is invertible
Let $\mathbb{F}_q$ be a finite field. Let $S_1, S_2, \cdots, S_m$ be $n \times n$ known matrices with entries in $\mathbb{F}_q$. Assume that each of $S_1, \cdots, S_m$ is invertible, and also $S_1, \...
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Why can a line search make an inexact Newton method perform more poorly on a linear system of equations?
Why can a line search make an inexact Newton method perform more poorly on a linear system of equations?
I have a problem where I need to solve multiple linear systems of equations where the system ...
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If a symmetric matrix $A$ is PSD, then $a_{ij} \leq \sqrt{a_{ii} a_{jj}}$ [duplicate]
In the 3rd lecture of Stephen Boyd's 2023 Stanford EE364A, one student mentions that in positive semidefinite matrices, every entry is less than or equal to the geometric mean of the diagonal entries ...
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Jacobian w.r.t. eigenvalues and eigenvectors of a (vectorized) matrix
Let $\mathbf{A} \in \mathbb{R}^{n \times n}$ be an real symmetric and invertible matrix such that its eigenvalues satisfy: $\lambda_1 > \lambda_2 > \cdots \lambda_n$. Moreover, let $\{\mathbf{u}...
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I don't understand the Jordan real form [closed]
My book, when it explains how to construct the jordan real form, they start by doing the complex endomorphism $\hat{f}$ of $\hat{V}$ with a matrix $A= \mathfrak{M}_\mathit{B}(\hat{f})$, they construct ...
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