All Questions
Tagged with convex-optimization linear-algebra
5 questions
6
votes
1
answer
4k
views
Complexity for solving linear equations?
What is the best known complexity for finding a vector $x \in \mathbb{R}^n$ to minimize $||Ax - b||^2$ and/or to solve (when possible) the system of linear equations $Ax=b$?
I am interested in ...
6
votes
2
answers
875
views
Is there any efficient solution of the matrix equation AXB + (AXB)' + cX = D
I am trying to find the symmetric solution $X\in \mathbb{R}^{p\times p}$ of following matrix equation:
$AXB + (AXB)^T + cX = D$
where $A,B\in \mathbb{R}^{p\times p}$ are symmetric positive ...
3
votes
1
answer
775
views
Is there some quantitative version of interlacing of eigenvalues of a matrix under rank-1 update?
Given a real symmetric matrix $A$ and a vector $v$ of the same dimension we know that the eigenvalues of $A + vv^T$ are left interlaced by the eigenvalues of $A$.
But do we have any quantitative ...
2
votes
1
answer
1k
views
Subgradient of Minimum Eigenvalue
Consider three $N \times N$ Hermitian matrices $A_0$, $A_1$, $A_2$. Consider the function
\begin{align}
f(t_1,t_2)=\lambda_{\text{min}}(A_0+t_1A_1+t_2A_2)
\end{align}
where $\lambda_{\text{min}}$ ...
2
votes
2
answers
2k
views
Hessian of function of covariance matrices
Suppose we have a typical logdet function $\mathcal{L}$ with respect to a covariance matrix $\mathbf{A}$,
$$
\mathcal{L}(\mathbf{A}) = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - \mathbf{q}^T(\...