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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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How to show $A$ cannot have more than one Jordan block for any eigenvalue?

I got stuck in this problem from Spring 99, Berkeley Problems in Mathematics: Let $A$ be a $n\times n$ matrix such that $a_{ij}\not=0$ if $i=j+1$ but $a_{ij}=0$ if $i\ge j+2$. Prove that $A$ cannot have more than one Jordan block for any…
Bombyx mori
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Find 10 commuting $2\times 2$ matrices of the same order

Prove that there exists 10 distinct real $2\times 2$ matrices which are pairwise commuting and all of the same finite order. Here, the order of matrix A is the smallest integer $k > 0$ such that $A^k = I.$ Also by 'pairwise commuting', I mean…
Merkh
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Eigenvalue of a matrix

Let $A$ be an $n\times n$ matrix and let $I$ be the $n\times n$ identity matrix. Show that if $A^{2} = I$, and $A \neq I$, then $\lambda =-1$ is an eigenvalue of $A$. This problem doesn't seem that too hard to solve, but I am stuck near the end.…
Kelly
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Properties for a matrix being invariant under rotation?

Consider a 2D case. Let $R$ be a rotation matrix with angle $\theta$ $$R = \begin{bmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{bmatrix}.$$ Is it possible for a matrix $A$ to satisfy the following identity for any $\theta$ $$A =…
newbie
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Can someone resolve my confusion about uniqueness of diagonalization?

I am a bit confused about diagonalization. I have $A$ which I know is diagonalizable. I want to find $P$ such that $A = P \Sigma P^{-1}$ where $\Sigma$ is diagonal. Under what circumstances is $P$ unique, if ever? If it is not unique, is it at least…
linearalgebra
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Linear algebra problem from dummite & foote

Let $V$ be a finite dimensional vector space over $\mathbb{Q}$ and suppose $T$ is a nonsingular linear transformation of $V$ such that $T^{-1} = T^2 + T$. Prove that the dimension of $V$ is divisible by $3$. If the dimension of $V$ is precisely $3$,…
Ethan Alwaise
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The projection operator on a finite dimensional vector space is diagonalizable

This question has been answered before, but I want to check if my solution using minimal polynomials is good. A projection matrix satisfies $M^2 = M$, so it satisfies the polynomial equation $M(M-1) = 0$. Thus the minimal polynomial must be either…
Schmidt
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Existence of polynomials in the Diagonal-nilpotent decomposition of a matrix

Let $V$ be a finite dimensional vector space over $\mathbb{C}$ and $T: V \rightarrow V$ linear operator. Show that there exist polynomials, without constant terms, $g, h \in \mathbb{C}$ such that $g(T)=D_T$ and $h(T)=N_T$, where $D_T$ is a…
user112358
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Proving dimension formula in linear algebra

Let $V$ and $W$ be finite dimensional vector spaces and let $T:V \to W$ be a linear transformation. (a) Prove that if $\dim(V) < \dim(W)$ then $T$ cannot be onto. (b) Prove that if $\dim(V) > \dim(W)$ then $T$ cannot be one-to-one. What I…
ys wong
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Understanding a proof of RREF uniqueness

Base Case $(n = 1)$: Suppose $A$ has only one column. If $A$ is the all zero matrix, it is row equivalent only to itself and is in reduced row echelon form. Every nonzero matrix with one column has a nonzero entry, and all such matrices have…
user312137
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Calculate complex determinant

$$\left| {\begin{array}{*{20}{c}}{{a^2}}&{{{(a + 1)}^2}}&{{{(a + 2)}^2}}&{{{(a + 3)}^2}}\\{{b^2}}&{{{(b + 1)}^2}}&{{{(b + 2)}^2}}&{{{(b + 3)}^2}}\\{{c^2}}&{{{(c + 1)}^2}}&{{{(c + 2)}^2}}&{{{(c + 3)}^2}}\\{{d^2}}&{{{(d + 1)}^2}}&{{{(d + 2)}^2}}&{{{(d…
Evgeny Semyonov
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Underdetermined Linear Systems

I'm working through an introductory linear algebra textbook and one exercise gives the system $2x+3y+5z+2w=0$ $-5x+6y-17z-3w=0$ $7x-4y+3z+13w=0$ And asks why, without doing any calculations, it has infinitely many solutions. Now, a previous exercise…
Liam Cooney
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About the uniqueness of rank-1 decomposition of a positive-definite Hermitian matrix

Suppose T is positive-definite Hermitian matrix and I know that it can be expressed by eigen-decomposition as the following sum of rank-1 matrices:$ \textbf{T}= \sum \lambda _{k} \textbf{u}_{k} \textbf{u}_{k}^{H} $where $\textbf{u}_{k} $ are…
cuiyi
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linear system solution, iterative vs direct

Dear all, I have systems like $(A - \lambda B) X = F$ where lambda is being updated inside a loop. I also have a limited number of eigenvectors of the matrix pair (A, B), say 40 eigenpair from a previous analysis step. I could get the results with…
Umut Tabak
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What is the span of $(1, 1, 1), (0, -1, 1), (0, 0, -1) \in \mathbb R^3$?

What is the span of $(1, 1, 1), (0, -1, 1), (0, 0, -1) \in \mathbb R^3$? Supposing we haven't covered linear in/dependence, can we solve the problem as done below? The span is a set of all systems: $$ \left\{ \begin{array}{c} a+b\cdot 0 + c\cdot…
integer
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