Tag

linear algebra

2 posts

The de Bruijn–Springer theorem: the convex weights expressing the critical points in terms of the roots can be chosen with column sums equal to 1 - 1/n.

mathematicspolynomialslinear algebra+1 more

From Gauss–Lucas to de Bruijn–Springer

Gauss–Lucas puts the roots of the derivative inside the convex hull of the roots. De Bruijn and Springer conjectured in 1948 that they are pulled inward on average — it took 55 years and two proofs.

November 2, 20252 min read

Diagram illustrating the trace operation and its dual/tensor characterization

mathematics

mathematicslinear algebra

The Trace of a Matrix: A Dual and Tensor Perspective

A fresh look at the trace of a square matrix: its linearity, role in the dual space of operators, unique tensor characterization, and its connection to inner products and the transpose via the dual.

April 5, 20251 min read