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

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 […]

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.

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 […]

$$\begin{bmatrix}1 &1 &6\\4& 3& 2\\5 &2& 2\\5& 3& 4\\4& 2& 4\end{bmatrix}\begin{bmatrix}4\\5\\6\end{bmatrix} = \begin{bmatrix}3\\5\\4\\3\\2\end{bmatrix}. $$ 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 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?

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, […]

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 $\alpha$.

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?

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 […]

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