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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 ...
Michael's user avatar
  • 544
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 ...
John. Tang's user avatar
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 ...
user6818's user avatar
  • 1,893
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 $\lambda_{\text{min}}$ ...
dineshdileep's user avatar
  • 1,421
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(\...
liubenyuan's user avatar