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Results tagged with matrices
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user 121643
Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.
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Eigenvalues and eigenvectors of k-blocks matrix
The block matrix $X$ is a particular one. Set $\frac{n}{k}=m$; you may diagonalize $A$ or $B$ by the same unitary $U$, thus taking the diagonal block matrix $V$ with diagonal blocks $U$, the matrix $D …
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Accepted
Matrices with same eigenvalues
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' …
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Existence of a specific stochastic matrix
A sketch of proof is as follows:
First if $A$ and $B$ are two doubly stochastic matrices verifying $(\star)$ then $A\cdot B$ also verifies $(\star)$. … Up to a direct sum of identity matrices we take $y=\begin{pmatrix}r+h\\r\\\vdots\\r\\r-k\end{pmatrix}$ and $x=\begin{pmatrix}r+h-s\\r\\\vdots\\r\\r-k+s\end{pmatrix}$ and $h\ge s>0$, $k\ge s>0$, with $ …
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Transforming matrix to off-diagonal form
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 …
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Property of positive semi-definite
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 unitary diagonal matrix. …
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An inequality related to matrix trace
The inequality fails when we take non particular cases, for example non diagonal matrices. Say $U \Sigma U^T>0$, and $S=(U \Sigma U^T)^{-1}$. …
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A (linear) optimization problem subject to (linear) matrix inequality constraints
Up to a unitary matrix $U$, $X$ is diagonal with all eigenvalues between $0$ and $1/2$ the other constraint on $AX$ implies that the diagonal entries of $U^*AUU^*XU$ are $\le 0$ and so are those of $ …
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Accepted
Monotonicity of eigenvalues
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Using the well known determinant formula for block matrices with a commuting off-diagonal block, you obtain that the eigenvalues $\lambda$ satisfy $$\det\left(\begin{pmatrix}-\lambda I&F\\G&-\lambda …
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On the bounds of the sum of the squares of spectral variation of two real symmetric matrices
A bound seems $4n(n-1)$, attained for $A=J-I$ and $B=-A$ (as in the comment) -edit- if we allow any ordering of the eigenvalues. First $$\sum_{i=1}^n\lambda_i(A)^2=\text{Tr}(A^2)=\sum_{i,j}|a_{i,j}|^2 …
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Extend an inequality on matrix norms
On the space of $n\times n$ complex matrices $(\sum_{i=1}^k\sigma_i^p)^{\frac{1}{p}}$ for $k\le n$, $p\ge 1$ where $\sigma_i$ are the singular values arranged in decreasing order, is a unitarily invariant …
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