5
votes
1answer
122 views

Are there any known results on numerical ranges of rank-one positive semi-definite matrices?

In my problem, I came across numerical ranges of rank-one positive semidefinite matrices. Through Toeplitz-Hausdorff theorem and some other extensions, I know if there are at most three matrices, then ...
2
votes
1answer
135 views

derivative of sum of singular values

can someone point me to the direction how to calculate the derivatives of a sum of singular values of a matrix? I am trying to minimize $$\min_A \parallel A \parallel_*+ \cdots $$ where $\parallel A ...
1
vote
0answers
163 views

Incoherence of the row/column span

Due to V.Chandrasekaran., et al‎ (p.11) : In general for any $k$-dimensional subspace of $A_{n×n}$ we have that: $$\sqrt{(k/n)} \leq incoherence(A)\leq 1$$ where the lower bound is achieved (for ...
5
votes
3answers
358 views

Optimization problem on trace of rotated positive definite matrices

Given two $n \times n$ symmetric positive definite matrices $A$ and $B$, I am interested in solving the following optimization problem over $n \times n$ unitary matrices $R$: $$ \mathrm{arg}\max_R ...
2
votes
1answer
200 views

More than controlability: Speed of controllability!

Consider the continuous linear time-invariant system $$ \begin{array}{l} \dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t)\\ \mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t) ...
1
vote
0answers
84 views

Copositivity in matrix pencils

Given two square symmetric matrices $A,B$ of the same order, the matrix pencil $P(A,B)$ is the set of linear combinations of $A$ and $B$. Finsler's theorem gives an elegant criterion for $P(A,B)$ to ...
5
votes
2answers
248 views

Simultaneous maximization of two Generalized Rayleigh Ritz Ratios

Consider hermitian positive semi-definite matrices $A_1$ and $A_2$. Consider also positive definite matrices $B_1$ and $B_2$. I want to maximize the minimum of the two Generalized Rayleigh Ritz ratios ...