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Results tagged with matrix-theory
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user 22051
Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.
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Moore-Penrose question
Let $A=BD^{\dagger}B^{T}$. I am looking for conditions under which $A^{\dagger}$ is a "nice" expression in $B$ and $D$ and their Moore-Penrose pseudo-inverses.
Do you know of such conditions?
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'Condition number' for Rayleigh-Ritz quotient
Suppose that $A$ is a Hermitian matrix and that $u,v$ are two vectors. Is there some known function $\kappa(A)$ so that $||u-v|| \leq \kappa(A) |\frac{u^{\*}Au}{u^{\*}u}-\frac{v^{\*}Av}{v^{\*}v}|$?
U …
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matrices whose entries sum to zero
Let $A$ be a non-singular matrix and let $s(A)$ be the sum of its entries. Under which conditions can it be assured that $s(A) \neq 0$?
if you like, you can assume that $A$ is symmetric.
Here is an …
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Is the trace of a Lyapunov transform of a semistable matrix always nonpositive?
Let $A$ be a semistable real matrix (i.e. the real parts of all the eigenvalues of $A$ are nonnegative). Let $P$ be a positive definite matrix.
Is it always true that $\operatorname{trace}{A^{T}P+PA …
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eigenvector update formula
Suppose that $B$ is a Hermitian matrix with one known eigenpair $(\lambda,v)$. (assume its the smallest or largest pair, if you like). Form the rank one update $B+\rho bb^{T}$.
Now I'm interested in …
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sharper interlacing
The usual interlacing inequalities say that if $M$ is a Hermitian $n \times n$ matrix and $\hat{M}$ is a principal submatrix of order $n-1$, then $\lambda_{\min}(M) \leq \lambda_{\min}(\hat{M})$. I wo …
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S-matrix conjecture: status?
Is the $S$-matrix conjecture still open? I mean the one listed as Problem 7 in this survey.
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Eigenvector localizaiton
I have raised this sort of question before but I think that now I've found a better term for the subject, one which might ring more bells for people - hence the repost. Hope you won't be too angry wit …
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When are cones of matrices "generated" by vectors?
The cone $P_{n}$ of positive semidefinite matrices of order $n$ can be represented in this form: $P_{n}=\{A|\forall x\geq 0: \langle A,xx^{T}>0 \rangle \}$ with $x$ running over $\mathbf{R}^{n}-0$.
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inverse-closed matrix spaces
Is there a known characterization of such spaces?
An example: the space of $n \times n$ matrices spanned by $I$ and $J$ (the identity and all-ones matrices, respectively) is inverse closed by the Sh …
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concentration for eigenvectors
I am interested in bounding from above the ratio between the maximum and minimum entries of a Perron vector. The only results that I found in the literature are from the classic masters (Ostrowski and …
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bounds on the entries of an inverse circulant matrix
Suppose that $C$ is a (real) circulant invertible matrix defined by a vector $d$. Then $C^{-1}$ is also a circulant defined by some vector $f$. There exists a standard formula that expresses the entri …
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Ring-theoretic version of a matrix problem
Problem #17 in Zhan's survey of open problems in matrix theory is the Li-Poon problem on writing a square real matrix as the linear combination of $k$ orthogonal matrices. They proved that it is possi …
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Estimating a spectral gap
Suppose you have a real positive definite matrix $A$ who eigenvalues are $\lambda_{1} \leq \lambda_{2} \leq \ldots \leq \lambda_{n}$. I am interested in bounding from below $\lambda_{2}-\lambda_{1}$. …
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name for a matrix operation
If $A$ is a matrix and $D$ is a diagonal matrix, is there some special name for $DAD$?
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