| Version 4 |
Version 3 |
| Let $A=(a_{ij})$ with $1\leq i,j\leq n\in\mathbb{N}$ be a square matrix with complex coefficients. Then the following inequality holds: |
Let $A=(a_{ij})$ with $1\leq i,j\leq n\in\mathbb{N}$ be a square matrix with complex coefficients. Then the following inequality holds: |
| $$|\det(A)|\leq \prod_{i=1}^n\left(\sum_{j=1}^n|a_{ij}|^2\right)^\frac{1}{2}.$$ |
$$|\det(A)|\leq \prod_{i=1}^n\left(\sum_{j=1}^n|a_{ij}|^2\right)^\frac{1}{2}.$$ |
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Moreover, if $A$ is Hermitian and positive semi-definite, the following inequality holds:
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Moreover, if $A$ is hermitian and positively semi-definite, the following inequality holds:
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| $$\det(A)\leq \prod_{i=1}^n a_{ii},$$ |
$$\det(A)\leq \prod_{i=1}^n a_{ii},$$ |
| with equality if and only if $A$ is diagonal. |
with equality if and only if $A$ is diagonal. |
