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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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Prove $A$ positive definite $\Rightarrow$ $A$ invertible

At the demonstration of this, I couldn't understand why the following holds: "$A$ positive definite $\Rightarrow$ $A$ invertible, because otherwise would exist $X\not=0$ satisfying $AX=0\Rightarrow X^TAX=0$ wich is a contradiction." I understood the…
Marcelo
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Coset and set of all cosets

Let $C$ be a subspace of a vector space $V$ and $x \in V$. The set $x + C = \{ x + c : c \in C \}$ is called a coset of $C$. The set of all cosets is denoted $V/C$. Can you help me with some examples and intuition behind the concept of coset? I…
xNJL
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find matrix representation of projection onto kernel

How can I efficiently find a matrix $P_A$ that projects onto the kernel of another matrix $A$? That is, given $Ax=0$, is there an "efficient" (without using inverses) way to find $P_A$ such that $P_A x \in ker(A)$, i.e., $A(P_A y) = 0$ for $Ay \neq…
jjjjjj
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Prove: $\operatorname{rank} A \leq 1$ iff $A=xy^T$ for some $x,y \in \mathbb{R}^{n \times 1}$.

Let $A \in \mathbb{R}^{n \times n}$. Then $\operatorname{rank} A \leq 1$ if and only if $A = xy^T$ for some $x, y \in \mathbb{R}^{n \times 1}$. Need help. I can't find sufficient facts nor theorems to support my proof on the statement.
coshMix
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Is there a faster way to do this? Find an orthogonal matrix $P$ and a diagonal matrix $D$ such that $A=PDP^T$

Let $A$=\begin{pmatrix} 0 & 1 & 1\\ 1 & 0 & 1\\ 1 & 1 & 0 \end{pmatrix} Find an orthogonal matrix $P$ and a diagonal matrix $D$ such that $A=PDP^T$. (Hint: The eigenvalues of $A$ are all integers. I managed to do this by finding the eigenvalues…
Yellow Skies
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Numerical Linear Algebra

If $X$ is an appropriate inverse of the nonsingular matrix $A\in \mathbb C^{n\times n}$ then two different measures of the quality of $X$ are $\|AX - I\|$ and $\|XA - I\|$. What is the largest factor by which these two quantities can differ ? Thanks…
user50017
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If $Q$ is an orthogonal, then is $Q+\frac{1}{2}I$ invertible?

If $Q$ is an orthogonal matrix, then the matrix has orthonormal columns. I asked this question to my friend and he says: Let $Q= -\frac{1}{2}I$, then it is orthogonal, and $Q+\frac{1}{2}I$ is zero, so not invertible. But I think if we set…
pascl
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If $T^2 = I$, how one can proof that $V = W \oplus U$?

Let $K$ be a field with cardinality different of $2$ and let $V$ be a $K-$vector space. Let $T: V \to V$ be a linear operator such that $T^2 = I$. Let $W = \{ v \in V: \, Tv = v \}$ and $U = \{ v \in V : Tv = -v \}$. I'm trying to proof that $V = W…
user 242964
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Is a finite order endomorphism a rotation?

Let $V$ be a real $2$-dimensional vector space, and $T\colon V\to V$ be an endomorphism such that $$ T^q = Id \qquad \textrm{and} \qquad T^j\not= Id\quad\textrm{if}\ 0
Alessio Di Lorenzo
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Eigenspaces of a symmetric matrix and its principal submatrices

There are a lot of well known theorems that relate the eigenvalues of a real symmetric matrix $A$ to the eigenvalues of its principal submatrices. I can't help but wonder, are there similar theorems which relate the eigenspaces in these cases? For…
J. Doe
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Codimension of intersection

Suppose $E$ is a vector space over a field of characteristic $0$. Let $E_1, F_1$ be subspaces of finite codimension and let $E_2, F_2$ be their respective complements, i.e., $E = E_1 \oplus E_2 = F_1 \oplus…
Lázaro Albuquerque
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One distinct eigenvalue $\iff A-\lambda E_n$ is nilpotent

Let $A\in \mathbb{C}^{n\times n}$ and $\lambda \in \mathbb{C}.$ $\lambda$ only eigenvalue of $A \iff A-\lambda E_n$ is nilpotent. So the first direction: let $\lambda$ be the only eigenvalue of $A$ then the characteristic polynomial of $A$ has the…
user463026
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Is $A^TA$ positive semi-definite for any real matrix $A$?

The question is written in title. I read a theorem saying: Suppose $A\in \mathbb R^{n\times n}$ is symmetric. Then the following are equivalent. $A$ is positive semidefinite. Eigenvalues of $A$ are all non-negative. $A$ can be factored as $A=G^TG$…
Maggie Mak
  • 339
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Find the kernel of a linear transformation of $P_2$ to $P_1$

For some reason, this particular problem is throwing me off: Find the kernel of the linear transformation: $T: P_2 \rightarrow P_1$ $T(a_0+a_1x+a_2x^2)=a_1+2a_2x$ Since the kernel is the set of all vectors in $V$ that satisfy $T(\vec{v})=\vec{0}$,…
Tim
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Antisymmetric matrix operating on $\mathbb R_{\ge 0}^n$

While looking at something related to game theory, I came across this problem. Given an antisymmetric matrix $\mathbf A$, show that there is a vector $\mathbf t \ne \mathbf 0$ with only nonnegative entries such that $\mathbf{At}$ has only…
eyeballfrog
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