Comment by Rodrigo de Azevedo on Evaluation of the determinant of a...
Do you agree with my edits?
View ArticleComment by Rodrigo de Azevedo on How to show the two convex bodies are...
The convex body $A$ is a spectrahedron known as spectraplex. At least some of its boundary is the set of rank-1 projection matrices.
View ArticleComment by Rodrigo de Azevedo on How to show the two convex bodies are...
The 1st link is linking to your MO profile. Is this desired?
View ArticleComment by Rodrigo de Azevedo on An approach to numerical mathematics using...
You might find the following Notices of the AMS interesting: †Braverman & Cook's Computing over the reals: foundations for scientific computing (2006) †Blum's Computing over the reals: where...
View ArticleComment by Rodrigo de Azevedo on Condition number can be arbitrarily worse...
Goes to infinity as what goes to infinity? Where is the convex optimization?
View ArticleAnswer by Rodrigo de Azevedo for Least-norm solution of the matrix equation...
Given symmetric matrix $\bf A$ and invertible square matrix $\bf B$, we have the following least-norm problem in square matrix $\bf X$$$\begin{array}{ll} \underset{{\bf X}}{\text{minimize}} & \|...
View ArticleOn computing the condition number of SPD matrices via convex optimization
Suppose that we have an $n \times n$ symmetric positive definite (SPD) matrix $\bf Q$ and that we would like to compute its condition number via convex optimization. In section 3.2 of Boyd et...
View ArticleAnswer by Rodrigo de Azevedo for An inequality for certain positive-definite...
$$ \tilde{\bf G} := \begin{bmatrix} 1 & \,\,\, {\bf a}^\top \\ {\bf a} & {\bf G} \end{bmatrix} $$where ${\bf G} \succ {\bf O}$ is a correlation matrix and ${\bf a} \geq {\bf 0}$. Since ${\bf G}...
View ArticleAnswer by Rodrigo de Azevedo for Bounding the norm of the inverse of a...
Given the invertible matrix ${\bf A} \in {\Bbb R}^{n \times n}$, we have the linear system ${\bf A} {\bf x} = {\bf 1}_n$, whose (unique) solution is denoted by ${\bf x}_0 := {\bf A}^{-1} {\bf 1}_n$. We...
View ArticleComment by Rodrigo de Azevedo on Unbiased estimator of the singular values of...
You might want to take a look at Alex Gittens's Random methods for Linear Algebra (2009) [PDF]
View ArticleComment by Rodrigo de Azevedo on Slight skew in the distribution of...
Normally distributed means $\sim \mathcal N (0,1)$?
View ArticleComment by Rodrigo de Azevedo on How can the dynamic programming be used to...
Do you agree with my edits?
View ArticleComment by Rodrigo de Azevedo on How do the singular values of a Hankel...
Where does this problem come from?
View ArticleAnswer by Rodrigo de Azevedo for INVERT transform as determinants of certain...
Let the HessenbergToeplitz matrix-valued function ${\bf M}_n : {\Bbb R}^n \to {\Bbb R}^{n \times n}$ be defined by$$ {\bf M}_n ({\bf a}) := \begin{bmatrix} a_1 & a_2 & a_3 & a_4 & a_5...
View ArticleComment by Rodrigo de Azevedo on Matching matrix columns under scaling,...
Do you know how to put the centroid of the columns of, say, $\bf Q$ at the origin? Right-multiply it by the rank-$(n-1)$ projection matrix ${\bf I}_n - \frac1n {\bf 1}_n {\bf 1}_n^\top$
View ArticleComment by Rodrigo de Azevedo on Designing a loss function in the form of a...
Do you agree with my edits?
View ArticleComment by Rodrigo de Azevedo on On the expected quality of a rank-$1$...
Instead of worrying about the phase, how about optimizing over $x_i \in {\Bbb C}$ subject to $x_i^* x_i = 1$ or $x_i x_i^* = 1$?
View Article