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For two positive vectors $a,b$ such that $a\prec b$, we know that there is an $m$ sequence of vectors $c^{(i)}$ such that $$a\prec c^{(1)}\prec \ldots \prec c^{(m)}\prec b$$ where each vector in the …

I just want to add a proof that may be given independently: Suppose we are given $k$ mutually orthogonal vectors in ${\mathbb{R}}^n\backslash\{0\}$; in matrix form this is a $k\times n$ matrix $A=[a_ …

The inequality fails when we take non particular cases, for example non diagonal matrices. Say $U \Sigma U^T>0$, and $S=(U \Sigma U^T)^{-1}$. The inequality is equivalent to prove $$\text{Tr}(SB^T|B|) …

To answer say a previous question, call such tridiagonal matrix $T$; the matrices $T$ and $-T$ (of dimension $n$) are unitarily congruent (they have the same spectra), that is there is an 'alternating …

This could be proven as follows (if i understood). Given a complex block matrix $M\in\mathbb{M}_m(\mathbb{M}_d)$ of dimension $n$ as $d\times m=n$. Assume that all diagonal entries of $M$ are not zero …

I am just putting a long possible comment here. The following has many reformulations; the idea is to give a condition for which the two matrices $A$ and $M$ in the OP question are congruent by a unit …

A bound seems $4n(n-1)$, attained for $A=J-I$ and $B=-A$ (as in the comment) -edit- if we allow any ordering of the eigenvalues. First $$\sum_{i=1}^n\lambda_i(A)^2=\text{Tr}(A^2)=\sum_{i,j}|a_{i,j}|^2 …

The $C$ is a particular block matrix, $C\in \mathbb{M}_3(\mathbb{M}_2(\mathbb{C}))$. For $V$ unitary let $V\begin{pmatrix}0&a\\\bar a&0\end{pmatrix}V^*=\begin{pmatrix}s&0\\0&-s\end{pmatrix}$, $P$ the …

I got to understand say $n=2$ and $s=d$, you are putting $\langle x_1;v_i\rangle=\langle x_2;v_i\rangle$ for all $i=1,\ldots, d$. This implies by linear combination, as $(v_1;\ldots;v_d)$ is a basis, …

For fixed $N$ the answer is yes, the case $\lambda=0$ should be easy, for $\lambda\neq 0$ one can take the candidate eigenvector $v$ whose first entry is zero $v=(0;x_1;x_2;\ldots;x_N)$ $(x_1=1)$ and …

The idea is to apply a unitary congruence $$U=\dfrac{1}{\sqrt{2}}\begin{pmatrix}I&-I\\I&I\end{pmatrix}.$$ I consider here $\mathcal{B}$ to be hermitian, $\mathcal{B}=\mathcal{B}^*$, whereas the genera …

Suppose we have $A_i$, $i=1\ldots n$, $n\times n$ complex matrices linearly independent. It may be conjectured that there exist $(a_1,\ldots,a_n) \in \mathbb{C}^n$ not all zero such that $\sum_{i=1}^ …

I came across the following while doing some related proof; It seems easy to prove. $\quad$ We are in ${\mathbb{M}}_n(\mathbb{C})$, $n>1$: $1$) Given a unitary $n\times n$ matrix $U$, there is some …

I came across the following conjecture, reading a recent paper in the Monthly, an orthogonal matrix of order $n\neq 0 \pmod 4$ has a nonnegative (up to a scalar) row vector. It should be straight in …