Analogue of the Cauchy-Schwarz inequality for determinants: a simple proof
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929294954987520 |
|---|---|
| author | Sidi, Avram |
| author_facet | Sidi, Avram |
| contents | In this note, we present a simple proof of an analogue of the Cauchy-Schwarz inequality relevant to products of determinants. Specifically, we show that $$ |\det(A^*MB)|^2\leq \det(A^*MA)\cdot \det(B^*MB),\quad A,B\in \mathbb{C}^{m\times n},$$ where $M\in\mathbb{C}^{m\times m}$ is hermitian positive definite. Here $m$ and $n$ are arbitrary. In case $m\leq n$, equality holds trivially. Equality holds when $m>n$ and $\text{rank}(A)=\text{rank}(B)=n$ if and only if the columns of $A$ and the columns of $B$ span the same subspace of $\mathbb{C}^m$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19691 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analogue of the Cauchy-Schwarz inequality for determinants: a simple proof Sidi, Avram General Mathematics 15 In this note, we present a simple proof of an analogue of the Cauchy-Schwarz inequality relevant to products of determinants. Specifically, we show that $$ |\det(A^*MB)|^2\leq \det(A^*MA)\cdot \det(B^*MB),\quad A,B\in \mathbb{C}^{m\times n},$$ where $M\in\mathbb{C}^{m\times m}$ is hermitian positive definite. Here $m$ and $n$ are arbitrary. In case $m\leq n$, equality holds trivially. Equality holds when $m>n$ and $\text{rank}(A)=\text{rank}(B)=n$ if and only if the columns of $A$ and the columns of $B$ span the same subspace of $\mathbb{C}^m$. |
| title | Analogue of the Cauchy-Schwarz inequality for determinants: a simple proof |
| topic | General Mathematics 15 |
| url | https://arxiv.org/abs/2403.19691 |