Articles of matrices

How can I prove that a matrix is area-preserving?

How can I prove that a $2\times2$ matrix $A$ is area-preserving iff $\det(A)=1$ or $\det(A)=-1$?

Prove that $A+B=-2I_{4}$

If $A,B\in M_{4}\left( \mathbb{R} \right) :A\neq B;Tr\left( A\right) \neq 0$ and $$\left\{\begin{matrix} A^2 – 2B + I_4 = 0_4 \\ B^2 – 2A + I_4 = 0_4 \end{matrix}\right.$$ prove that: $$A+B=-2I_{4}$$ and $$\det\left( A-aI_{4}\right) \geq \det\left( A+aI_{4}\right) ,\forall a\in \mathbb{R} $$ All my ideas ($A^{2}=2B-I_{4};B^{2}=2A-I_{4}\Rightarrow A^{2}B=2B^{2}-B;B^{2}A=2A^{2}-A$) seem to be wrong as long as I can’t […]

Matrix Identity

Let $A$ be a $ n \times n $ positive definite matrix, $P$ be a $n\times m$ real matrix with full column rank, I’m wondering whether we have the following inequalty $$ P( P^{T} A P) ^{-1} P^{T} \preceq A ^{-1} ,$$ And in which sufficient conditions the the equality holds.

Simulating simultaneous rotation of an object about a fixed origin given limited resources.

Sorry if the title is a bit cryptic. It’s the best I could come up with. First of all, this question is related to another question I posted here, but that question wasn’t posed correctly and ended up generating responses that may be helpful for some people who stumble upon the question, but don’t address […]

Matrix multiplication using Galois field

$$\begin{bmatrix}1 &1 &6\\4& 3& 2\\5 &2& 2\\5& 3& 4\\4& 2& 4\end{bmatrix}\begin{bmatrix}4\\5\\6\end{bmatrix} = \begin{bmatrix}3\\5\\4\\3\\2\end{bmatrix}. $$ I am not getting that how come this result is possible ? [Editor’s comment #1: The question makes sense, but the asker forgot to explain their notation – possibly because they have not been exposed to any alternatives (happens regrettably often […]

$J$ be a $3\times 3$ matrix with all entries $1$ Then $J$ is

$J$ be a $3\times 3$ matrix with all entries $1$ Then $J$ is Diagonalizable Positive semidefinite $0,3$ are only eigenvalues of $J$ Is positive definite $J$ has minimal polynomial $x(x-3)=0$ so 1, 2,3 are true , am I right?

For a matrix $A$, is $\|A\| \leq {\lambda}^{1/2}$ true?

In class I saw a proof that went something along these lines: Define $\|A\| = \sup \dfrac{\|Av\|}{\|v\|}$ for v in V, where the norm used is the standard (Does this even exist?) Euclidean norm in V. $\|Av\|^2 = <Av, Av> = <A^TAv,v>$ where $<,>$ denotes a dot product. Note that $A^TA$ is a non-negative matrix, […]

Proof of Non-Convexity

Am looking for a proof of non-convexity of the quotient of two matrix trace functions as given by $\frac{\operatorname{Tr}X^TAX}{\operatorname{Tr}X^TBX}$, when $TrX^TBX>0$ for two different positive semi-definite matrices , $A$ and $B$ such that $A \neq \alpha B$ for any scalar $\alpha$.

Upper triangular matrix and nilpotent

How to show that an upper triangular matrix is nilpotent if and only if all its diagonal elements are equal to zero by using induction?

Can any linear transformation be represented by a matrix?

Use $\cal L$ to denote a linear transformation on some vector space. We know any matrix $\bf{A}$ can be viewed as a linear transformation by defining $\cal L:= \cal L(\bf{v})= Av$ where $\bf{v}$ is a vector. I am curious is any linear transformation can be represented by a matrix? If so, why? If not, can […]