Articles of linear algebra

$A \in Gl(n,K)$ if and only if $A$ is a product of elementary matrices.

I want to show the statement: $A \in GL(n,K)$ if and only if $A$ is a product of elementary matrices. I could show the reverse implication: Suppose $A=T_1…T_m$ where each $T_k$ is an elementary matrix. An elementary matrix can be of the form: 1) $T_{ij}$ the matrix that swaps the rows $i,j$ 2) $T_{\lambda i}$ […]

Determinant of skew- symmetric matrix

I have to prove that determinant of skew- symmetric matrix of odd order is zero and also that its adjoint doesnt exist. I am sorry if the question is duplicate or already exists.I am not getting any start.I study in Class 11 so please give the proof accordingly. Thanks!

Total Derivative and Multilinear Functions

So I’m starting to work through Spivak’s Calculus on Manifolds and I’m having a little trouble verifying some of the claims made in the book problems. To review: Given a function $\mathbf{f}:\mathbb{R}^{n}\to\mathbb{R}^m$, we say that $\mathbf{f}$ is differentiable at a point $\mathbf{a}=(a^{1},\ldots,a^{n})\in\mathbb{R}^n$ (considered as a $1\times n$ matrix) if there exists a linear transformation, $D\mathbf{f}(\mathbf{a}):\mathbb{R}^n\to\mathbb{R}^m$, […]

The form of the states on an algebra of $n\times n$ matrices with complex entries

I have a question concerning $n \times n$ matrices. Denote by $M_n(\mathbb{C})$ the algebra of all $n \times n$ matrices with complex entries. Let $\phi$ be a state on $M_n(\mathbb{C})$ i.e. a linear functional $\phi \colon M_n(\mathbb{C}) \rightarrow \mathbb{C}$ such that $\phi(Id)=1$ and $\phi(A^*A) \geq 0$, for each $A \in M_n(\mathbb{C})$. Show that $\phi$ must […]

Asymptotes and focus of a conic?

This is the conic $$x^2+6xy+y^2+2x+y+\frac{1}{2}=0$$ the matrices associated with the conic are: $$ A’=\left(\begin{array}{cccc} \frac{1}{2} & 1 & \frac{1}{2} \\ 1 & 1 & 3 \\ \frac{1}{2} & 3 & 1 \end{array}\right), $$ $$ A=\left(\begin{array}{cccc} 1 & 3 \\ 3 & 1 \end{array}\right), $$ His characteristic polynomial is: $p_A(\lambda) = \lambda^2-2\lambda-8$ A has eigenvalues discordant […]

Prove two sets span the same subspace

I’ve found here that in order for two sets to span the same subspace, the following must be true: Each vector in S1 can be written as a linear combination of the vectors in S2; and Each vector in S2 can be written as a linear combination of the vectors in S1. I don’t know […]

A property of incidence matrix of a graph

Let $G$ be an oriented graph with incidence matrix $Q$, and let $B:=[b_{ij}]$ be a $k\times k$ sub-matrix of $Q$ which is non-singular. Can there exist two distinct permutations $\sigma$ and $\sigma^\prime$ of $1,\ldots ,k$ for which both the products $b_{1\sigma (1)}\cdots b_{k\sigma (k)}$ and $ b_{1\sigma^\prime (1)}\cdots b_{k\sigma^\prime (k)}$ are non-zero ?

I cannot make the mental leap from a vector to a function!

In my linear algebra book, it says that a vector is linearly independent if $\vec V = c1*\vec T_1 + c2*\vec T_2$ And then it goes on to say that $y(t) = c1 * e^{-at} + c2*e^{-bt}$ is linearly independent My mind cannot comprehend how an analogy can be made in this case. Is there […]

Proof that multilinear rank of tensor less of equal than border rank

Let’s consider the tensor $t \in \mathbb{R}^{d_1 \times d_2 \times d_3}$. It is well-know statement, that multilinear rank of tensor less or equal than border rank, which means that if we consider multilinear rank $rk_{mult} = (rk_1, rk_2, rk_3)$ than $rk_i(t) \le brk(t)$. I have the proof of this simple fact, but I can not […]

Does there exist a unique “max” approximator for matrices?

In engineering sometimes a limit of a $p$-norm, (or in practice for some conveniently large $p$): $$\underset{x_k \text{ s.t. }x_k>x_j \forall j}{\underbrace{{\max({\bf x})}}} \approx \underset{p\to \infty}{\lim}\|{\bf x}\|_p=\underset{p\to \infty}{\lim}\left(\sqrt[p]{\sum_{\forall i} ({x_i})^p}\right)$$ is used as a continous approximation to the “max”-function, finding an approximation to the largest scalar $x_k$ in $\bf x$. Could we find some natural […]