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Assume that $V$ is a finite dimensional real vector space of dimension $n$. Is there a $\mathbb{R} -$ valued $k$- linear map $T$ on $V$ which is not an alternative form but it vanish on all $k$- tuple …

Is there a non trivial sequence $(T_{n})$ of linear operators $T_{n}$ on $M_{n}(\mathbb{C})$ such that $$T_{nm}(X\otimes Y)=T_{n}(X) \otimes T_{m}(Y) $$ where $X$ and $Y$ are in $M_{n}(\mathbb{C})$ …

According to the answer of znt to the previous version, I revise the question as follows: Is there a real $(n-1)\times n$ matrix $A$ such that $A$ is not a full rank matrix and satisfy $a_{ii} …

For $n>2$, are there norms $\parallel.\parallel_{a}$ and $\parallel.\parallel_{b}$ on $M_{n}(\mathbb{R})$ with the following property: $A\in M_{n}(\mathbb{R})$ is singular if and only if $\para …

Let $S\subset M_{n}(\mathbb{R})$ be the singular points of the equation $Det=0$. That is $S$ is the critical points of the determinant function. What matrices belongs to $S$, precisely? Let $M= …

Let $(A_{1},A_{2}, \ldots,A_{k})$ be $k$ matrices in $M_{n}(\mathbb{R})$. Is there an algebraic formula, as a generalization of "Determinant" for $k=1$, to compute the joint spectrum of …

I think that the part $(a)$ of proposition $2.4$ of this paper shows that for $n$ sufficiently large one can construct a $n \times n$ unitary matrix $U=-I_{2}\oplus U'$, which can be decomp …

One can consider an alternative concept of prime matrix as follows: A matrix $A\in M_n(\mathbb{Z})$ is prime if for any factorization $A=BC$ we have either $Det(B)\in \{-1,1\}$ or $Det(C)\in \{-1,1 …

How can one compute each of the following matrices, explicitly: $$\int_{O(n)} e^{g}dg$$ or $$\int_{O(n)} g^{n}dg \;\;\;\;n\in \mathbb{N} \;\;n>1$$ What is the explicite entries of the resulting ma …

I am considering the previous version of your question which contained two parts. Part 1) We assume $Q<P$ which means that $P-Q$ is a strictly positive matrix. Of course this condition is a …

Assume that $X$ is a vector field on a $n$ dimensional manifold $M$.Let $0\leq i,j \leq n$. 1.Is there a linear isomorphism $T:\Omega^i(M) \to \Omega^j(M)$ with $L_X T=T L_X$? 2.Is there a linear i …

Every matrix $A\in M_4(\mathbb{R})$ can be represented in the form of $$A=\sum_{i=1}^{n(A)} B_i\otimes C_i$$ for $B_i,C_i\in M_2(\mathbb{R})$. What is the least uniform upper bound $M$ for such $n(A) …

Is there a vector field $X$ on $\operatorname{M}_n(\mathbb{R})$ or $\operatorname{GL}(n,\mathbb{R})$ with the following condition: $$\begin{cases} X\cdot \operatorname{trace}=\operatorname{Det} \\X\c …

Is it true to say that every matrix $A\in M_n(\mathbb{R})$ is similar (conjugate) to a matrix $B=(b_{ij})$ with $b_{ij}=-b_{ji}$ for all $i\neq j$?(With some abuse of terminology,a matrix $B$ with th …

According to the interesting answer of Rado, "trace" is an algebraic way to represent the dimension of fibres of a vector bundle on a compact Hausdorff space $X$. Every vector bundle on $X$ c …