English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

A theorem of Mirsky provides necessary and sufficient conditions for the existence of an N -square complex matrix with prescribed diagonal entries and prescribed eigenvalues. A simple inductive proof of this theorem is given. Theorems of Alfred Horn (3) and Leon Mirsky (4) give necessary and sufficient conditions for the existence of an N -square matrix A that has prescribed diagonal elements and prescribed eigenvalues. In the case of Horn's Theorem, A is required to be Hermitian, and in the case of Mirsky's theorem no constraint is imposed on A. While simple inductive proofs of Horn's Theorem have been published, the only published proof of Mirsky's Theorem that we are aware of is the original, which is neither simple nor transparent. In this note, we give a simple proof of Mirsky's Theorem which we relate to Chan and Li's proof (2) of Horn's Theorem. We begin by recalling the results in (3, 4) in more detail.

Short proofs of theorems of Mirsky and Horn on diagonals and eigenvalues of matrices