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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.

Linear algebra is concerned with vector spaces and linear transformations between them:

$$(x_1, \dots, x_n)\to a_1x_n+\dots+a_nx_n$$

Concepts include systems of linear equations, bases, dimensions, subspaces, matrices, determinants, kernels, null spaces, column spaces, traces, eigenvalues and eigenvectors, diagonalization and Jordan normal forms.

This is a general tag, most of the subjects included have secondary tags, e.g.

127034 questions
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Uniqueness of sum of nilpotent and diagonalizable matrices.

I have the following question: Let $V$ be a vector space over a field $F$, and let $A$ be an endomorphism $V\rightarrow V$. Prove there is at most one pair of linear maps $D$ and $N$ such that $D+N=A$, $D$ is diagonalizable, $N$ is nilpotent, and…
Dorothy Kimmel
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Is sub-vector an established mathematical entity?

Reading the following paragraph I was wondering how this entity that the authors [1] call a sub-vector should be named. In Matlab a sub-vector has to be contiguous. Let $\mathbf{X}=(X_1,\ldots,X_n)$ be a vector of random variables and…
Anne van Rossum
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Prove that if matrix $A$ is nilpotent, then $I+A$ is invertible.

So my friend and I are working on this and here is what we have so far. We want to show that $\exists \, B$ s.t. $(I+A)B = I$. We considered the fact that $I - A^k = I$ for some positive $k$. Now, if $B = (I-A+A^2-A^3+ \cdots -A^{k-1})$, then…
J Park
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When $\operatorname{span} (S_1 \cap S_2) = \operatorname{span}(S_1) \cap \operatorname{span}(S_2)$holds

Let $S_1$ and $S_2$ be subsets of a vector space. When does the equality $$\operatorname{span} (S_1 \cap S_2) = \operatorname{span}(S_1) \cap \operatorname{span}(S_2)$$ hold? I have found two sufficient conditions: They are vector spaces; one is a…
student
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Does $A$ admit a square root with integer entries?

Let us consider the matrix $$ A = \left( \begin{array}{crc} 0 & 0 & 3 \\ 81 & 0 & 0 \\ 0 & 3 & 0 \end{array} \right).$$ Does the matrix $A$ admit a square root each of whose entry is an integer? Please help me in this regard. Thank you very…
little o
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How do we know you can only flip something’s orientation two times?

I was exploring what the determinant’s sign means geometrically. For 2-D, you can swap the axes and you’ve flipped orientation, and once you swap them again, you get the original orientation. Why should this be the case for higher dimensions? Why is…
Yatharth Agarwal
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Understanding the distance between a line and a point in 3D space

I know that there are quicker ways to do what I am about to present. But I want to understand why my approach does not work. Let the point $P = (-6, 3, 3)$ and the line $L=(-2t,-6t,t)$. I am trying to find the shortest distance between the point and…
Cedric Martens
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If $A$ is a matrix and $p$ is a polynomial such that $p(A)=0$ then must the roots of $p$ necessarily be eigenvalues of $A$?

This is just a small query: If $A$ is an $n\times n$ square matrix and $p(t)=(t-\lambda_1)(t-\lambda_2)\cdots(t-\lambda_m)$ be a polynomial (with $\lambda_i \in \mathbb{C}$ for all $i=1, \ldots, m$) such that $p(A)=0$, then is it necessary that…
Sayantan
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What does calculating the inverse of a matrix mean?

Assume I have 3 equations, $x+2y+z=2, 3x+8y+z=12, 4y+z=2$ which could be represented in matrix form ($Ax = b$) like this: $\begin{pmatrix} 1 & 2 & 1\\ 3 & 8 & 1\\ 0 & 4 & 1 \end{pmatrix}\bigl .\begin{pmatrix} x\\ y\\ z \end{pmatrix} =…
Eyad H.
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A "geometric''-ish infinite sum of matrices

Suppose I have full rank $n\times n$ matrix $A$ with $\rho(A) < 1$ and I want to find an expression for $$S = X + A^\top X A + A^{2\top} X A^2 + A^{3\top} X A^3 + \dots$$ where $X$ is an $n\times n$ positive definite matrix. Thus, $$ S = X +…
Benjamin Christoffersen
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Simultaneous eigenvectors of symmetric and antisymmetric parts

This question will seem over-specific and obscure, but the motivation comes from a problem I am trying to solve in game theory. I hope someone can help, as it requires only linear algebra! Let $H$ be a real invertible matrix, decomposed into…
smalldog
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Singular Value Decomposition in terms of Change of Basis?

I do see how SVD can be understood in terms of rotating. But it is hard for me to understand SVD in terms of change of basis. So, let me start from Diagonalization to clarify my question. When $B=\{v_1, \dots, v_n\}$ where $v_i$ are eigenvectors,…
12345678
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Equivalence of Two Norm and Infinity Norm

How could you show that: $$\|x\|_\infty \le \|x\|_2 \le \sqrt{n} \|x\|_\infty. $$ I was able to show the left hand side but got stuck showing the right hand side. What would be the best way to approach it? For the LHS: $$\|x_j\|_\infty = \max|x_j|…
Is12Prime
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Find area of pentagon using determinants

Find the area of the pentagon of the five vertices $(1,2), (4,1), (5,3), (3,7), (2,6)$ . Please, use the way of using determinant. My idea is to cut the pentagon into some triangles, then calculate each triangle, then sum them. I wonder if there…
user53800
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Linear transformation of a polygon maximizing its area with respect to its perimeter.

Given a polygon $P$ on the plane, is there a rigorous method or algorithm to compute or approximate a linear transformation $T$ which maximizes the following ratio? $$\frac{\mathrm{Area}[T(P)]}{\mathrm{Perimeter}[T(P)]^2}$$
wircho
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