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Results tagged with matrix-theory
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user 23141
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.
3
votes
Accepted
Fixed point of quantum operations
As clarified in the comments, the question refers to positive operator valued measures, not quantum operations. Basically the issue is whether, for a given $A \in M_n = M_n(\mathbb{C})$ the map $B \ma …
- 42.8k
13
votes
Accepted
Differentiability of operator norm
It need not be differentiable everywhere. Let $P$ and $Q$ be mutually orthogonal self-adjoint projections. Then the norm of $P+ t Q$ is 1 for $|t| \leq 1$ and $|t|$ for $|t| > 1$.
However, $\|A + tB\ …
- 42.8k
5
votes
Accepted
What is the significance of matrix ordered algebras?
Okay. What you have defined is a matrix ordered vector space, not a matrix ordered operator algebra. But this is actually a better place to start.
First question, give examples of such things. The st …
- 42.8k
3
votes
A property of positive matrices
No, this is false. Consider $\left[\matrix{1&0&0&1\cr 0&0&0&0\cr 0&0&0&0\cr 1&0&0&1}\right]$, regarded as a $2 \times 2$ matrix of $2\times 2$ blocks. This is a rank 1 matrix with operator norm 2 but …
- 42.8k