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Results tagged with matrix-analysis
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user 18060
The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.
2
votes
A matrix diagonalization problem
The general $n=m$ case: First take $X$ to be the identity, this gives us that $W$ is the diagonal. Then for each pair of entries on the diagonal, look at the set of matrices that are almost entirely t …
- 149k
8
votes
Accepted
Factor matrix ${\bf A}$ into the product ${\bf B}{\bf C}$ where ${\bf C}$ has no negative en...
Let $v_1, \dots, v_n$ denote the rows of $A$.
Let $u$ be a vector with all positive entries that is not a linear combination of $v_2, \dots, v_n$. Then we may write $v_1 = c u + a_2 v_2 + \dots a_n …
- 149k
5
votes
Accepted
Smallest singular value of $X\mapsto AX^{T}+XA^{T}$
There is always a zero singular value as soon as $n \geq 2$.
Write $A = UD V$ with $D$ diagonal and $U, V$ orthogonal. Then we can write $X \mapsto AX^T + X A^T$ as $$X \mapsto UDV X^T + X V^T D^T U^ …
- 149k
2
votes
Accepted
Singularity of matrix pencil-like expression
No.
The first condition is satisfied if (and only if) there is some vector in the kernel of $A$ that is also in the kernel of $B$.
The second condition is satisfied (if and) only if the kernel of $A$ …
- 149k
1
vote
Eigenvalues of an amplification matrix
Let $C=A+B$ be any real matrix. Let $A$ be any real matrix, then $B=C-A$ is a real matrix. Then $e^{i\theta} A+ B = (e^{i\theta}-1) A + C$. Since $A$ can be any real matrix, this bears essentially no …
- 149k
2
votes
upper bounds on a certain matrix norm
Probably not unless $A$ and $B$ are positive-definite, since if $B$ is very close to $-A$ then $B^{-1}+A^{-1}$ is very small and so its inverse is very large. In fact, depending on the norm, they prob …
- 149k
5
votes
Accepted
Representation of $4\times4$ matrices in the form of $\sum B_i\otimes C_i$
Because $M_4(\mathbb R) = M_2(\mathbb R) \otimes M_2(\mathbb R)$ as vector spaces (and as algebras, but we won't use this), we can replace $M_2(\mathbb R)$ by an arbitrary $4$-dimensional vector space …
- 149k
8
votes
Accepted
M-matrix plus S-matrix is P-matrix?
This is not true in general.
$\left(\begin{array}{ccc} 4 & 0 & -16 \\ 2 & 4 & 0 \\ 0 & 2 & 4 \end{array}\right)= \left(\begin{array}{ccc} 1 & -2 & -16 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{array}\right …
- 149k