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        Questions tagged [matrices]

        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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        14 views

        How can i determine the elements of this matrix?

        i am new to matrices, i am learning with a book but i am stuck at this task, i need to determine the elements of this matrix: and this is the solution : i dont understand how to get there, any help ...
        0
        votes
        0answers
        38 views

        Vector Inequality Transformation based on Chebyshev ball (centered at the origin) [on hold]

        Consider $y$, $x_1$ and $x_2$ are vectors that satisfy the following inequality: $$ x_1 \leq y \le x_2 $$ according the above inequality, I think there should be exist a relation such as: $$ y^T ...
        4
        votes
        0answers
        205 views

        Do we know what the impulse to “introduce” the Jordan canonical form was?

        Mo-ers, Do you know how it was that the study of the Jordan canonical form began? There are certain things that may be said once one has thought about the matter: for instance, one can say that the ...
        4
        votes
        1answer
        149 views

        Trace of inverse of random positive-definite matrix in high dimension?

        Consider a random matrix $A \in \mathbb{R}^{n\times n}$ with i.i.d. entries, with symmetric law and finite variance. I am curious about the behavior of $$\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1}...
        3
        votes
        0answers
        49 views

        How can I find the integral orthogonal group of a given symmetric positive definite form?

        I wonder how one can study the integral orthogonal group of a given (symmetric, positive definite) bilinear form like the one described by the following matrix: $$M=\begin{bmatrix} x_1 &...
        3
        votes
        0answers
        69 views

        Algebra of block matrices with scalar diagonals

        I am interested in block matrices $A$, that is $A\in M_{n\times n}(R)$ where $R=M_{s\times s}(k)$ and $k$ is a field, such that for every positive integer $m$ the matrix $A^m$ has only scalar blocks ...
        4
        votes
        1answer
        78 views

        Lipschitz property of matrix function only depending on singular values

        Let $f$ be a function from $\mathbb{R}^{n\times n}$ to $\mathbb{R}$ such that there exists another symmetric function $g$ (invariant under permutation of coordinates) from $\mathbb{R}^{n}$ to $\mathbb{...
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        57 views

        How to obtain mathematical expectation with the vector as random variable?

        In my study, I wish to get the mathematical expectation for the term below. The vector $\boldsymbol{z} \in \mathcal{C}^{N\times1}$ and $\boldsymbol z \sim \mathcal{CN}\left(\boldsymbol{0},\boldsymbol{...
        2
        votes
        0answers
        63 views

        Generalization of Cauchy's eigenvalue interlacing theorem?

        Cauchy's Interlacing Theorem says that given an $n \times n$ symmetric matrix $A$, let $B$ be an $(n-1) \times (n-1)$ principal submatrix of it, then the eigenvalues of $A$ and those of $B$ interlace. ...
        1
        vote
        1answer
        124 views

        How can one integrate over the unit cube, subject to certain (quantum-information-theoretic) constraints?

        To begin, we have two constraints \begin{equation} C1=x>0\land z>0\land y>0\land x+2 y+3 z<1 \end{equation} and \begin{equation} C2=x>0\land y>0\land x+2 y+3 z<1\land x^2+x (3 z-2 ...
        1
        vote
        1answer
        277 views

        Recovering “$n$” from $M_n(\mathbb{C})$

        Is there an example of an infinite-dimensional $C^*$-algebra $A$ which admits the following structure: The $C^*$ algebra $A$ admits a faithful trace $tr$ such that the multiplication $m: A\otimes A \...
        7
        votes
        2answers
        477 views

        Does a non-singular matrix have a large minor with disjoint rows and columns and full rank?

        Say I have an $n$-by-$n$ non-singular matrix $A$ all of whose diagonal entries are $0$. We call an $m$-by-$m$ minor of $A$ good if its set $I$ of row indices and its set $J$ of column indices ($I,J\...
        3
        votes
        1answer
        152 views

        Representation of $4\times4$ matrices in the form of $\sum B_i\otimes C_i$

        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 $n(A)$...
        2
        votes
        1answer
        46 views

        Equalities between transforms of matrices that are extremely different

        I have two $2N\times 2N$ matrices, defined by blocks: $$ A = \begin{bmatrix} a & 0 \\ 0 & 0 \end{bmatrix} $$ $$ B = \begin{bmatrix} 0 & 0 \\ 0 & b \end{bmatrix} $$ where $a$ and $b$ ...
        3
        votes
        1answer
        379 views

        Eigenvalue-taking operator?

        $\newcommand{Tr}{\operatorname{Tr}}$ Is there a continuous map $(p,t) \mapsto \lambda(p,t)$ which, given a path $p: [0,1] \to M(2,\mathbb R)$ and a $t \in \mathbb [0,1]$, gives back an eigenvalue of $...

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