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Results tagged with matrices
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user 9022
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
Ways to convert a Positive Semi-Definite (PSD) matrix -> Positive Definite matrix
The choice that you make can result in a huge difference in the solution. Neither method is particularly good and both can be quite unstable.
For example, suppose that
$M=[1\;0\;0\; 0;\; 0 \; …
- 3,934
1
vote
Accepted
Understanding the derivation of a ML-estimator (statistics)
You can simply multiply $\Omega^{-1}$ as given by the author times $\Omega$ and see that the product is $I$. This confirms that the formula for $\Omega^{-1}$ is correct.
The key here is that
$(I- …
- 3,934
2
votes
Cholesky Rank-1 downdate extension
Computing $L_{*}^{-1}$ from $L_{*}$ takes only $O(n^2)$ time.
If you can afford the two matrix-matrix multiplications (which are $O(n^3)$ but parallelize and use cache very efficiently), then that …
- 3,934
4
votes
Trace of multiplied positive definite matrices
You also haven't given any indication of the size of the matrices involved. … For this answer, I'll assume that the matrices are dense and have no other easily exploited special structure, and that the matrices are large enough that asymptotic worst case analysis of running time …
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3
votes
Accepted
Is there any conclusions generalized Singular Value Decomposition into Hilbert Space
The simplest generalization is that a "compact self adjoint linear operator" on a Hilbert space can be diagonalized in terms of (an infinite number of) eigenvalues and eigenfunctions that are elements …
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4
votes
How to approx. decompose a sym. p.d. matrix M into X'X?
Unfortunately, you can't do any better than your "no so good method." It's a standard result that this is the best (as measured by the Frobenius norm) rank $n$ approximation to $M$.
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3
votes
Optimizing directly on the eigenspectrum of a matrix
Many (but not all) problems involving the eigenvalues of a graph are convex optimization problems that can be formulated as semidefinite programming problems. There are a number of "tricks" that you …
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5
votes
Accepted
How do I optimize over (or take derivative wrt) a square diagonal matrix?
Your notation is somewhat confusing, in that you apply the subscript $i$ to $w$, and have a vector $w_{i}$, but don't use $i$ in any meaningful way in your problem. I'm going to take the liberty of …
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7
votes
Accepted
Explicit formula for Cholesky factorization in a special case
The matrix $\alpha J$ is a rank one matrix, so there are simple update/downdate formulas for computing the Choleksy factorization of $Q+sI-\alpha J$ if you start with the factorization of $Q+sI$.
I …
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5
votes
How to solve Ax=b incrementally ?
As Fumiyo Eda already mentioned, you can use an iterative method such as GMRES to resolve the system after the change to $A$.
If you want to use direct LU factorization rather than an iterative meth …
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6
votes
efficient way to compute the inversion of the following matrix
As others have already indicated, there's a good chance that you don't actually need $(E-aX)^{-1}$, but rather $(E-aX)^{-1}v$ for some vector $v$. In that case, you might be better off using an itera …
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9
votes
sparsity of QR decomposition
The Givens rotations are transformations that are extremely sparse/structured matrices. …
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