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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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Gap between the induced norm of a matrix and largest Eigenvalue?

Are there any known results on how big the gap between the absolute value of the largest Eigen value of matrix and the induced norm can be? More formally, let the induced norm of A is given by $\|A\| = max_{\|x\| = 1}\|Ax\|$ and let $\lambda_{max}$…
Thiagarajan
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Is having the sum of each column zero a sufficient condition for a matrix to be singular?

If I have a square matrix, $$ \begin{pmatrix} a_{1,1} & \ldots & a_{1,n} \\ \vdots & \ddots & \vdots \\ a_{n,1} & \ldots & a_{n,n} \\ \end{pmatrix} $$ Is a sufficient condition for the matrix to have…
Vincent Tjeng
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Equation of a line that passes halfway between two points (in other words, divides the space)

Is there a formal proper way of finding the line between two points? By that I don't mean the line connecting the two points, I mean a line that runs the same distance away from point 1 and point 2. To phrase it another way, I want to find the…
CodyBugstein
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Must eigenvector matrix be invertible?

When reading eigenvector of a matrix, there is a formula: $AP = PD$ where in $P$, each column is A's eigenvector and $D$ is diagonal matrix with diagonal element being A's eigen values. Now coming the question: Is matrix P always invertible?…
user2262504
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A basis of a subspace is subset of a basis of the whole space

If $X$ is a vector space with a basis $B$ and $A$ is a subspace of $X$. Does $A $always has a basis subset of $B$? If yes, how should I prove this? If no, we should give an example of a vector space $X$ with a basis $B$ and a subspace $A$ of $X$…
Sara
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Finding basis of $R(T)$

Let $T: \text{Mat}_{2 \times 3} \rightarrow \text{Mat}_{2 \times 2}$ be defined by $$T \left(\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix}\right) = \begin{pmatrix} a_{11} + a_{12} & a_{12} + a_{13} \\ a_{21} +…
user144806
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Change of basis?

So the question is... A transformation $T$ is denoted by $T(x,y)=(x+y,x-y)$. $C$ is the basis $\{(1,-1),(1,1)\}$ $D$ is the basis $\{(1,2),(1,0)\}$ I know $T(C)=\{(0,2),(2,0)\}$ But how do I express that in terms of $D$ and more importantly what…
user139985
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These inner products don't match in $\mathbb C^n$

In $\mathbb C^n$, we can define the inner product between $u=\{u_1,\ldots,u_n\}$ and $v=\{v_1,\ldots,v_n\}$ as $\langle u,v\rangle=u_1\overline{v_1}+\ldots+u_n\overline v_n$. I've read in a book that we can define in $\mathbb C^n$ the inner product…
user42912
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Does $u v^T + v u^T$ have exactly one positive and one negative eigenvalue when $u \not \propto v$?

Does $u v^T + v u^T$ have exactly one positive and one negative eigenvalue when $u \not \propto v$? $u$ and $v$ are column vectors in $\mathbb{R}^n$.
opt
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Show that $A + A^{-1} \geq 2I$ for $A > 0$.

For a positive matrix $A$. Here, we assume that all positive matrices are self-adjoint. Show that \begin{align} A + A^{-1} \geq 2I. \end{align} Here, $A≥0$ means that A is self-adjoint and for all $x∈ℂn,⟨Ax,x⟩≥0.$
DRich
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Difference between tuple and row matrix

In Munkres Analysis on Manifolds the author uses the word ''tuple spaces'' to refer to a special sort of vector space. Further down on the same page (page 6) he discusses the linear isomorphism that maps a tuple to a row matrix. I am greatly…
blue
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orthogonal complement of a sum

$A, B$ subspaces of $V$, a finite-dimensional inner product space. SHOW [note oc = orthogonal complement (which is defined below)] 1) $oc(A+B) = oc(A) \cap oc(B)$ 2) $oc(A \cap B) = oc(A) + oc(B)$
user128616
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Negative eigenvalue of doubly stochastic nonnegative matrices

I am studying n by n doubly-stochastic entry-wise positive matrices. I was wondering if there are any necessary and sufficient conditions for the existence of a negative eigenvalue for such a matrix. What if I added that the matrix was also…
Giovanni
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Prove condition number of a invertible matrix is atleast one

Show that the condition number of an invertible matrix must be at least 1. What matrices have condition number equal to 1. If someone could help me with this and give an explanation that would be very helpful. I do not know where to start
Vogtster
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Find a change in variable that will reduce the quadratic form to a sum of squares

Find a change of variable that will reduce the quadratic form $x_1^2-x_3^2-4x_1x_2+4x_2x_3$ to a sum of squares, and express the quadratic form in terms of the new variable.
user0692
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