Mathematics Stack Exchange
Mathematics Stack Exchange

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
5
votes
1 answer

Proving that $||A-B||=||A+B||\Leftrightarrow AB=0$

I have to prove that \begin{equation*} ||A-B||=||A+B||\Leftrightarrow AB=0 \end{equation*} and I was wondering if this approach is correct, or if there's a better/more elegant way to prove this. Given n-dimensional vectors A and B, we can write…
Fernando Martin
  • 5,847
5
votes
1 answer

Linear functional on matrix space, nonnegative on positive semidefinite matrices

Let $f:M_n(\mathbb C) \to \mathbb C$ be a linear function such that $f(x^* x)\ge0$ for all $x$ and $f(1)=1$. Show that there exist $\alpha_1,...,\alpha_k\in \mathbb C^n$ such that $f(x)=\sum_{i=1}^{k}\langle x\alpha_i,\alpha_i \rangle$ for all…
TIMP
  • 159
5
votes
4 answers

sum of the eigenvalues = trace($A$)?

Is it true that for a square matrix $A$, all of whose eigenvalues exist in the base field, sum of the eigenvalues = trace($A$)? The result holds in all the matrices I've studied.
Sriti Mallick
  • 6,137
  • 3
  • 30
  • 64
5
votes
0 answers

Finding all $T$-invariant subspaces of a $T$-cyclic vector space

Let $V$ be a $4$-dimensional real vector space and let $T:V \rightarrow V$ be an endomorphism such that its characteristic polynomial $\chi_T$ is given by $\chi_T = (t-2)^3t$. Suppose also that $V$ is $T$-cylic, that is to say, $V = \{p(T)(v_0):…
user2345678
  • 2,885
5
votes
3 answers

what value of K does the system have a unique solution

$\begin{cases}x_1 + kx_2 - x_3 = 2\\2x_1 - x_2 + kx_3 = 5\\x_1 + 10x_2 -6x_3= 1\\ \end{cases}$ I've been trying echelon form where i took $R_2 = R_2 - 2R_1$ and $R_3 = R_3-R_1$ So I have…
MatrhWonder00
  • 69
5
votes
3 answers

About restriction of linear map

Let $A$ be a linear map from the vector space $X$ to $Y$ and $T$ be a subspace of $X$ . I want to understand what is the meaning of saying that the restriction $A_{|T}: \to A(T)$ is invertible. Could anybody explain me what is this restriction…
mathscrazy
  • 1,837
5
votes
1 answer

calculate kernels of matrices with angles

So my professor gave me this question: I have to find the basis of the eigenvalues of this matrix \begin{pmatrix} \cos(q) & \sin(q)\\ \sin(q) & -\cos(q)\\ \end{pmatrix} so I calculate the eigenvalues and I found it is 1 and -1. so now I need…
wantToLearn
  • 1,275
5
votes
1 answer

Consequences of a rectangular matrix being of maximal rank

I have a real matrix $A$, $(m+1) \times m$ and a vector $b \in \mathbb R^{m+1}$ such that $b_{m+1}=0$. For any vector $u\in \mathbb R^m$, $Au=0 \Rightarrow u=0$. This means that $A$ is a rectangular matrix of maximal rank, i.e. of rank $m$. Since…
JKH
  • 141
5
votes
3 answers

Orthogonal complement of the diagonal matrices in the inner product space of matrices

$V$ is the matrices space (scalar over the complex). definition of inner product space is: $(A,B)=tr(AB^*)$. $A$,$B$ matrices. assuming $D$ is the subspace of all Diagonal matrices. I need to find the subspace that each matrix $B$ in it $(A,B)=0$.…
wantToLearn
  • 1,275
5
votes
2 answers

The orthogonal complement of the space of row-null and column-null matrices

I propose the following lemma and its proof. It is related to row-null and column-null matrices - i.e. matrices whose rows and columns both sum to zero. Could you please give your opinion on the plausibility of the lemma, and the validity of the…
D R
  • 1,158
5
votes
3 answers

Problem related with a linear transformation and its rank,nullity...

I am stuck on the following problem: Suppose $W$ be a vector space of dimension $m \geq 2.$ Let $F \colon W \to W$ be a linear transformation such that $F^{n+1}=0$ and $F^n \ne 0$ for some $n \ge 1.$ Then I have to check which of the following…
user53386
5
votes
3 answers

Least Singular Value of bidiagonal matrix

Consider the $n \times n$, bidiagonal matrix $$ \left(\begin{array}{ccccc} x & \\ 1 & x \\ & 1& x \\ & & & \ddots \\ & & & 1 & x \end{array} \right) $$ It is claimed that the least singular value for this matrix is less than $O(2^{-n})$ when $x$ is…
JohnKnoxV
  • 1,015
5
votes
2 answers

Does Real Eigenvalues mean it is an hermitian Matrix

Let us say I know that a given $N\times N$ matrix has all its eigenvalues as real, does it mean, it is hermitian. How do I prove (or disprove) that?
dineshdileep
  • 8,887
5
votes
2 answers

Spanning a Vector space of matrices by symmetric and skew symmetric matrices.

How do I span a vector space of $4\times 4$ matrices with real values by symmetric and skew symmetric matrices? The basis of vector space of $4\times 4$ matrices has 16 elements, each containing one 1 and fifteen 0's. All I have to figure out is…
user67773
5
votes
1 answer

A problem on linear algebra

If $M$ is a $3 \times 3$ matrix such that $$[ 0 ~~1 ~~2 ]M = [ 1 ~~0~~ 0 ] \text{ and } [ 3~~ 4 ~~5 ]M = [ 0 ~~1 ~~0 ]$$ then what is the value of $[ 6 ~~7 ~~8 ]M$ ? I guess some matrix property needs to be used.
user21982
1 2 3
…
99 100