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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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A matrix $M$ that commutes with any matrix is of the form $M=\alpha I$

I feel like this is probably a simple proof but I can't quite come up with it in an elegant way nor could I find it here. Prove that if a matrix $M$ commutes with any matrix then $M$ is of the form $M=\alpha I$. Proving the contrapositive seems like…
kyp4
  • 720
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Eigenvalue of $F^2$ and$ F$

Let $F$ be a linear operator on vector space $V$ over $\mathbb{C}$ and let $\lambda \in \mathbb{C}$ Show that if $\lambda^2$ is eigenvalue of linear operator $F^2$ then at least one of $\lambda,-\lambda$ is eigenvalue of $F$. What I tried to do was…
Mateusz
  • 852
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Invertibility of uncertain matrix

Given that we start with some $n\times n$ square matrix $A_0$ that is non-singular. We add some perturbation to it, $\Delta A$, so that our new matrix is $A = A_0 + \Delta A$. The question is whether we can guarantee invertibility of our new $A$…
user130912
  • 132
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Dimension of the range of $T$ is equal to the codimension of $\ker T$

Given a linear transformation, $T:U\rightarrow V$ , I am asked to show that the dimension of the range of $T$ is the same as the codimension of the kernel of $T$. I am told that $U$ is not necessarily a finite dimensional vector space so I cannot…
aponga
  • 63
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2 answers

SVD-like matrix decomposition based on any basis?

Let's say I have a point $\mathbf{x}$ in $n$-dimensional space. For any basis $(\mathbf{u}_1, ..., \mathbf{u}_n)$, $\mathbf{x}$ can be written as a linear combination of this basis. $\mathbf{x} = x_1 \mathbf{u}_1 + x_2 \mathbf{u}_2 + ... + x_n…
Federico Magallanez
  • 1,878
6
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Average Line of a Set of Lines

Suppose we have 10 lines in an x-y plane. The lines are somewhat clustered together, and going more or less in the same direction. The data I have for these lines is their line equation: $$y = a + bx$$ I'm wondering how one can come up with an…
nkormanik
  • 161
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Why is this formula positive definite?

I have a formula $$A(I+GQ)^{-1}(G+GQG)(I+QG)^{-1}A^{\mathrm T}+G$$ where $A,Q,G,I\in\mathbb R^{n\times n}$, $A$ nonsingular, $G$ positive semi-definite, $Q$ positive definite, $I$ the identity matrix, and $(A,G)$ controllable. It is obvious that…
user143763
  • 433
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3 answers

A gap in Halmos' definition of dimension? And how to repair?

In Halmos' Finite-Dimensional Vector Spaces, section I.8 has a proof of the Steinitz exchange lemma, which says that if $V$ is a vector space, $S$ is a finite independent subset of $V$, and $T$ is a finite generating subset of $V$, then the…
nham
  • 93
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Dimension of a vector space

My question is: Is the vector space containing all periodic complex sequences a finite-dimensional vector space?
Chon
  • 6,002
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1 answer

Taking powers of a triangular matrix?

So there is a formula for the $n$th power of a matrix in Jordan normal form. Is there a formula for the $n$th power of a general triangular matrix? If not, are there known formulas for "nice" upper triangular matrices? Like those consisting of all…
Xavier
  • 99
6
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2 answers

If a matrix is upper-triangular, does its diagonal contain all the eigenvalues? If so, why?

If a matrix is upper-triangular, does its diagonal contain its eigenvalues? If yes, how can this be proven? My textbook and teacher just jumped over this statement (we are working over complex numbers, does the answer change if it's over reals?) and…
upptrang
  • 63
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$A$ is skew hermitian, prove $e^A$ is unitary

Given $A$ is a skew-hermitian, ( i.e $A^H = -A$ ) then how do you prove the matrix exponential $e^A$ is unitary. To prove the unitary property of the matrix, I need to show $(e^A)^{*}(e^A) = (e^A)(e^A)^{*}= I$. Can any one help me how to proceed and…
Learner
  • 2,696
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Invariant subspaces and eigenvectors

Let $T:V \rightarrow V$ be a linear transformation where $V$ is a finite dimensional vector space. Let $W \subset V$ be such that $T(W) \subset W$. If $v_1, v_2, \ldots ,v_n \in V$ are eigenvectors corresponding to distinct eigenvalues $\lambda_1,…
crypton480
  • 740
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Find an invertible matrix $P$ and a diagonal matrix $D$ such that $D=P^{−1}AP$?

I have a matrix $A=\begin{bmatrix} -5 & -1 & 2\\ 2 & 0 & -2\\ -6 & -1 & 3\end{bmatrix}$, and I need to find an invertible matrix P and a diagonal matrix D such that $D = P^{-1}AP$. I've found the eigenvalues for the matrix and they are $-3, 1, 0$,…
Johnathon Pomenrantz
  • 119
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1 answer

Finding a matrix with respect to a basis

Let $T: R^2 \to R^2$ be represented by $\begin{bmatrix}5 & -3\\2 & -2\end{bmatrix}$ with respect to the standard basis. Find the matrix T with respect to the basis B = { $\begin{bmatrix}3 \\1\end{bmatrix}$ , $\begin{bmatrix}1\\2 \end{bmatrix}$ }. I…
user127273
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