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Results tagged with matrices
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user 27249
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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Is this matrix positive semi-definite? [closed]
Consider $K$ vectors $x_1,\dots,x_K$ in $\mathbb{R}^N$. Define the $K\times K$ matrix $A$ whose $(i,j)$ entry is given as $$A_{ij}=\exp(-\frac{||x_i-x_j||^2}{2})$$ Is this matrix Positive Semi-Definit …
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3
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0
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182
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Relating Numerical Range and Perron-frobenius theorem for positive matrices?
Let $A$ be any matrix with all entries positive (which means Perron-Frobenius theorem can be applied). Then its numerical range is defined as the set of complex numbers
$$W(A)=\{x^HAx\lvert ~x^Hx=1\}$ …
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6
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2
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629
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Rank-one positive decomposition for a entry-wise positive positive definite matrix
I have asked this question in math.se without any success.
Let $\mathbf{A}$ be a symmetric $n\times n$ positive semi-definite matrix and also such that each of its entries is positive. Does $\mathbf{A …
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Mixing Numerical Range and inner product
Let $\mathbf{A}$ and $\mathbf{b}$ be a symmetric $N\times N$ real matrix and $N\times 1$ real vector respectively. Then consider the set of points in $\mathbb{R}^2$ defined as
\begin{align}
\mathbb{S} …
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483
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Is there any connection between this matrices
Matrices I discuss are all $N\times N$ hermitian matrices. Define two positive (semi)definite matrices $H_1$ and $H_2$. … Even special cases are welcome, for instance, say they are rank-one matrices, Does it make any difference? …
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170
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How does scaling rows to sum to 1, of a positive matrix change the perron vector?
Reposting from math.sx due to lack of response. Let $A$ be a $N\times N$ positive matrix such that $A_{ij}>0$. By Perron-Frobenius theorem, there is a unique positive left eigenvector called Perron ve …
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An Interesting variant of Rayleigh Quotient
Let $A$ and $B$ be two given hermitian positive semi-definite matrices, then what is the solution for
$$\max_{x\neq 0}\frac{x^HAx}{x^HBx+1}.$$
I am looking for closed form solutions. …
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2
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2
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282
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Inequality on numerical range of inverse of kernel matrix
Let $k(.,.)$ be a function that takes two vectors as input and outputs a scalar as follows
\begin{align}
\mathcal{k}(x,y) = \exp\left(-\frac{\|x-y\|_2^2}{2}\right)
\end{align}
where $\|x\|_2$ denotes …
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Proof for a rank-one decomposition theorem of positive (semi) definite matrices
$\DeclareMathOperator\rank{rank}\DeclareMathOperator\trace{trace}$Consider the following result which I recently came across in a research paper in my area (signal processing)
Let $X$ be a $N\times N …
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2
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294
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Rayleight Ritz Ratio and smallest eigenvalue for a set of given matrices
I am familiar with Rayleigh Ritz Ratio for hermitian matrices. Let $A_1$ be a given $N \times N$ hermitian matrix. … Any comments on what happens when I try it for $k$ matrices $A_1,A_2,\ldots,A_k$ …
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3
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308
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4
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1k
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Minimum eigenvalue of a Affine Combination of two Hermitian matrices
Consider two $N \times N$ hermitian indefinite matrices $A_1$ and $A_2$. …
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272
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A certain type of quadratic problem.
.~ u^{H}A_2u=0
\end{align}
$A_1$ and $A_2$ are $N\times N$ hermitian matrices. $u$ is the unit-norm $N\times 1$ complex vector I need to find. I have worked on it a bit and I am reaching no where. …
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Feasibility of a given set of homogenuous nonconvex quadratic inequality constraints
$C_N$ be $M \times M$ indefinite hermitian matrices. …
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Relating a Polynomial equation to the characteristic equation of a Hermitian matrix
This question arose out of mere curiosity. Given a polynomial equation and I happen to know that its roots are real (but not the roots itself). Does it mean it is the characteristic equation of a Herm …
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