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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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Almost projections in matrix algebras.

I have a question about projection in matrix algebras over the complex number, that I can not solve.. Let p be a matrix in some $M_n(\mathbb C)$ and suppose that p is almost self-adjoint (i.e. $||p-p^*||<\epsilon$) and almost idempotent (i.e.…
Alessandro Vignati
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Elementary Row/Column Operations and Change of Basis

Let $V$ and $W$ be finite-dimensional vector spaces and let $T:V \rightarrow W$ be a linear transformation between them. I have read that Performing an elementary row operation on the matrix that represents $T$ is equivalent to performing a…
ItsNotObvious
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Isometries of $\mathbb{R}^2$

Show that if $A:\mathbb{R}^2\to \mathbb{R}^2$ is a proper rotation, then it may be represented by a matrix of the form $$\pmatrix{ \cos(\theta)& -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \\}.$$ Further, any improper rotation is given by…
Lays
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Is matrix diagonalization unique?

From the following statement, it seems matrix diagonalization is just eigen decomposition. Diagonalizing a matrix is also equivalent to finding the matrix's eigenvalues, which turn out to be precisely the entries of the diagonalized matrix.…
WishingFish
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The maximum eigenvalue of the sum of two matrix

Suppose there are two matrix $A$ and $B$. The components of each matrix is non-negative. And $$Ax_1=\lambda_1 x_1 $$ where $\lambda_1$ is the maximum eigenvalue of $A$. Similarly $$Bx_2=\lambda_2 x_2 $$ where $\lambda_2$ is the maximum eigenvalue…
Vivian
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Applications of systems of linear equations

Sorry if this questions is overly simplistic. It's just something I haven't been able to figure out. I've been reading through quite a few linear algebra books and have gone through the various methods of solving linear systems of equations, in…
miggety
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difference between eigenspace and generalized eigenspace

What are the differences between eigenspace and generalized eigenspace? Why do we need generalized eigenspace? Can an arbitrary matrix (not necessarily over $\mathbb{C}$) have a Jordan form? Thank you very much.
LJR
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Finite sum of eigenspaces (with distinct eigenvalues) is a direct sum

I'm trying to show that a finite sum of eigenspaces (with distinct eigenvalues) is a direct sum. I have $ \alpha : V \to V $. The eigenspaces are $ V_{\lambda_i} = \ker(\alpha - \lambda_i id_V )$ for $ 1 \leq i \leq n $. My attempt at a proof: $ A…
user938272
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Prove that matrix is non-negative

Problem: Given $A_{1}, A_{2}, ..., A_{n}$ - finite sets and $a_{ij} = |A_{i}\cap A_{j}|$ - number of elements in intersection of sets. Prove, that matrix $(a_{ij})_{i=1,2,..,n}^{j=1,2,.., n}$ is non-negative. I've cleared out that this matrix is…
Ann Orlova
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Injective, surjective and bijective for linear maps

I have problems showing that $\phi$ is surjective. My understanding is, that I have to show for every $u \in \mathbb{R}^3$ that there exists a $v \in \mathbb{R}^3$ but I am not sure how. Let $a,b,c \in \mathbb{R}$. Let's examine the…
monoid
  • 608
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How do you orthogonally diagonalize the matrix?

How do you orthogonally diagonalize the matrix A? Matrix A = $$ \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix} $$
Kenneth Hend
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The path of complex structure.

$J = \left( {\begin{array}{*{20}{c}} 0&{ - {I_n}}\\{{I_n}}&0\end{array}} \right)$. If $I(t)$ is a path in $\rm M(2n, \mathbb R)$ ($2n \times 2n $ real matrix) such that $I(0)=0$ and $I(t)J+JI(t)=0$, then for sufficiently small $t$ can we find a path…
Totoro
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Is there a quick way to generate the characteristic polynomial of a Vandermonde matrix?

This came up on an exam recently as extra credit. The first part was to find the characteristic polynomial, $f_A = \text{det(}A - xI_n)$ where $I_n$ is the n by n identity matrix, of $A = \left| \begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2…
AvatarOfChronos
  • 1,701
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Primary decomposition problem

Let $T$ be a linear operator on a finite dimensional space $V$, and let $p=p_{1}^{r_{1}} \cdots p_{k}^{r_{k}} $ be the minimal polynomial for $T$, and let $V= W_{1} \oplus\cdots\oplus W_{k}$ be the primary decomposition for $T$, i.e., $W_{j}$ is the…
user71433
  • 101
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The calculation of $\dim(U + V + W)$

Is there a (valid) formula for $\dim(U + V + W)$? I know from MO that $$\begin{align*} \dim(U + V + W) &= \dim(U) + \dim(V) + \dim(W)\\ &\qquad\mathop{-} \dim(U \cap V) - \dim(U \cap W) - \dim(V \cap W)\\&\qquad \mathop{+} \dim(U \cap V \cap …
Vicfred
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