Articles of matrices

If both $A-\frac{1}{2}I$ and $A + \frac{1}{2}I$ are orthogonal matrices, then…

Problem : If both $A-\frac{1}{2}I$ and $A + \frac{1}{2}I$ are orthogonal matrices, then which one of the following is correct : (i) A is orthogonal (2) A is skew symmetric matrix of even order (3) $A^2 = \frac{3}{4}I$ Solution : $(A’-\frac{1}{2}I)(A-\frac{1}{2}I) =I$ and $(A’+\frac{1}{2}I)(A+\frac{1}{2}I) =I$ $\Rightarrow A +A’ =0$ $\Rightarrow A’ =-A $ $\Rightarrow A^2 […]

Anybody knows a proof of Uniqueness of the Reduced Echelon Form Theorem?

The book has no proof showing each matrix is row equivalent to one and only one reduced echelon matrix. Does anybody know how to prove this theorem? “Theorem Uniqueness of the Reduced Echelon Form Each matrix is row equivalent to one and only one reduced echelon matrix” Source: Linear Algebra and Its Applications, David, C. […]

Need help understanding matrix norm notation

I’ve been trying to understand matrix norms (full disclosure: school assignment, not looking for answers, just clarity!), and how they follow from vector norms – been awhile since I did much linear algebra, so i’m struggling a bit with the notation, in particular I’m solving in the general case that for matrix A (and any […]

Interpolation in $SO(3)$ : different approaches

I am studying rotations and in particular interpolation between 2 matrices $R_1,R_2 \in SO(3)$ which is: find a smooth path between the 2 matrices. I found some slides about it but not yet a good book, I asked the author of the slides and he told me he does not know about a good book […]

Differential and derivative of the trace of a matrix

If $X$ is a square matrix, obtain the differential and the derivative of the functions: $f(X) = \operatorname{tr}(X)$, $f(X) = \operatorname{tr}(X^2)$, $f(X) = \operatorname{tr}(X^p)$ ($p$ is a natural number). To find the differential I thought I could just find the differential of the compostion function first and then take the trace of that differential. Am […]

Does a matrix represent a bijection

We have a square binary matrix that represents a connection from rows to columns. Is there a way to tell if a bijection exists (other than checking for all possible bijections and iterating through them)? EXAMPLE: 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 1 ANSWER: Yes, here […]

Elementary proof that $Gl_n(\mathbb R)$ and $Gl_m(R)$ are homeomorphic iff $n=m$

This question already has an answer here: Elementary proof that $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}^m$ 6 answers

Hessian matrix for convexity of multidimensional function

To prove that a one dimensional differentiable function $f(x)$ is convex, it is quite obvious to see why we would check whether or not its second derivative is $>0$ or $<0.$ What is the intuition behind the claim that, if the Hessian $H$ of a multidimensional differentiable function $f(x_1,…,x_n)$ is positive semi-definite, it must be […]

What kind of matrix norm satisifies $\text {norm} (A*B)\leq \text {norm} (A)*\text {norm} (B)$ in which A is square?

$||A\times B||\le ||A||\cdot ||B||$ is not always correct. But which kind of matrix norm satisifies this formula for square matrix $A$ and arbitrary matrix $B$?

A is Mn×n(C) with rank r and m(t) is the minimal polynomial of A. Prove deg $m(t) \leq r+1$

$A$ is a matrix of $M_{n \times n}(\mathbb{C})$ with rank $r$ and $m(t)$ is the minimal polynomial of A. I need to prove that : deg $m(t) \leq r+1$ I need to find a condition of the matrix A, in which deg $m(t) = r+1$ Can anyone help me ? The solution involves the primary […]