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Results tagged with matrices
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user 5963
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.
4
votes
Accepted
Full rank submatrices of positive semidefinite matrix
Yes, this is true. To see this note that for $A$ positive semidefinite, $v^T A v = 0$ if and only if $Av = 0$. For the less obvious direction, write $A = B^TB$ for a real matrix $B$. Then $0 = v^TA …
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20
votes
Accepted
Prove that matrix is positive definite
Update: I originally claimed to prove that $A$ is strictly positive definite, but there was a bug in the strictness part. I have revised the proof to show that $A$ is positive semidefinite. For an e …
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2
votes
On the solvability of a matrix equation
Let $N=n$, $m=1$, $P_i = 1$ for all $i$, and let the $C_i$ be the standard unit vectors. Then the left hand side of $(\star\star)$ is the matrix whose diagonal entries are the inverses of the diagona …
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5
votes
Computational complexity of low rank SDP
Various NP-hard problems, such as MAX-CUT, can be formulated exactly as SDPs with rank constraints (just google e.g. "max cut sdp"). If you want to enforce such rank constraints anyway, a popular app …
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1
vote
Accepted
2
votes
Accepted
Space of matrices B for which there is a solution to Bx=c for a given c
If you are willing to replace the vectors $c$, $x$, and $y$ by the spaces $U_1 = \text{span}(c)$ and $U_2 = \text{span}(x,y)$, which in some sense doesn't change the problem, then one generalization i …
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3
votes
Accepted
Finding the most compact representation of a vector in an "overdetermined base"
This problem and various related problems are known to be NP-hard to solve exactly, but there has been a lot of work on efficient approximations. See this wikipedia page or try googling things like " …
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2
votes
Feasibility of a given set of homogenuous nonconvex quadratic inequality constraints
One option for a convex relaxation is to search for a positive semidefinite hermitian $W$ with $\mathrm{Trace}(WC_i)\geq 1$ for all $i$. These conditions are equivalent to the ones you have written w …
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4
votes
Accepted
Minimum norm solution of a least squares using SVD
Take a look at the wikipedia page on the Moore-Penrose pseudoinverse, specifically Sections 5.3, 6.1, and 6.3.
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0
votes
Eigenvalues of a sum of Hermitian positive definite circulant matrix and a positive diagonal...
If your constraints on the $d_i$ allow it and you are interested in numerical solutions, you can cast the problem as a semidefinite program. To do so, introduce scalar variables $t_1,\ldots, t_n$. L …
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3
votes
An inequality question
No. The inequality you have given implies (by scaling any $y\in V$) that $M^Ty=0$ for all $y\in V$. That is to say, $V\subseteq \operatorname{ker}\left(M^T\right)$. There is no such $V$ if $\operat …
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9
votes
Accepted
minimize the sum of absolute eigenvalues
So what you are saying is that you have an affine space of matrices (a "matrix pencil") over which you would like to minimize the nuclear norm. …
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