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Results tagged with matrices
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user 9833
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.
3
votes
Accepted
Maximising a Rayleigh quotient over a subspace
If $z=M^{-1/2}Qx$, then
$$K=\max_{z\in S}\frac{z^TMz}{z^Tz},$$
where $S$ is the range of $M^{-1/2}Q$.
- 6,891
1
vote
Accepted
Sufficient conditions for inverse-positivity
Sorry for promoting my own results,
but I think the condition in my old paper
"A sufficient condition for the monotonicity
of a positive definite matrix" (Computational Mathematics and Mathematical …
- 6,891
5
votes
Accepted
Is this lower bound for a norm of some complex matrices true?
No, it is not; in fact, $2(n-1)$ is a local maximum.
Let $B$ be a Hermitian matrix such that $|B_{ij}|=1$ and $B_{ii}=1$. We denote its eigenvalues by $\mu$ (not to confuse them with eigenvalues of $ …
- 6,891
1
vote
Accepted
Finding a particular matrix factor
This is impossible. Let
\begin{equation*}
C(x) = \left(
\begin{array}{cc}
a(x) & b(x) \\
c(x) & d(x)
\end{array} \right)
\end{equation*}
and $\theta(x)=\det C(x)$. From
$$a(x^{-1})d(x)+b(x)c(x^{- …
- 6,891
6
votes
Inverse of matrix $D + ADA^T$
To begin with, the matrix in question can well be degenerate, consider for example
\begin{equation*}
D=\left(
\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right),
A=\left(
\begin{array}{cc}
0 & -1 …
- 6,891
4
votes
Accepted
Square root of a large sparse symmetric positive definite matrix
I completely agree with fedja: there is a nice method here (which, unfortunately, does not always work well). If you know bounds for the spectrum of $A$, say $0<a<\lambda<b$, then you (sometimes) can …
- 6,891