# (Multi)linear Algebra

One of my favorite areas of mathematics.
I am especially fond of numerical (multi)linear algebra.

### A sufficient condition for non-negative solutions to non-negative linear systems

Here we are concerned with “non-negative” linear systems, that is, linear systems where $\mathbf{A}, \mathbf{b} \geq \mathbf{0}$ (elementwise). In particular, we give a sufficient condition for …

### The space of matrices where LU requires pivoting is almost full-dimensional

The pivoted LU factorization of square matrices is the key routine for the direct solution of linear systems. All square, invertible matrices have a pivoted LU factorization of the form $\mathbf{PA} = \mathbf{LU}$, …

### Behavior of conjugate gradient residual

An often stressed property of the conjugate gradient method (CG) for solving linear systems is the monotonic decrease in the A-norm of the error. When CG is applied in practice the exact solution is unknown and the error cannot be computed or tracked, so …

### Linear Algebra Pronunciation Guide

Like all fields of mathematics, linear algebra has many prominent figures whose names are non-trivial to pronounce in English (the lingua franca of science and mathematics).
This is further complicated by numerical linear algebra libraries which have inconsistent …

### Interior eigenvectors of symmetric matrices are saddle points

Eigenpairs of symmetric matrices are intimately related to optimization and critical points, with the eigenvectors being critical points of the Rayleigh quotient. In optimization settings, the type of critical point (minimum, maximum, …

### Relationship between power iteration and gradient descent

Although the majority of successful algorithms for the symmetric tensor eigenvalue problem use optimization techniques directly, there are a few notable algorithms that do not appear to be based on optimization. Rather, they more closely …