Hadamard's inequality

In mathematics , the Hadamard inequality describes an estimate for the determinant of a square matrix . It is named after the French mathematician Jacques Salomon Hadamard .

Classic Hadamard inequality

Let be a matrix over the complex numbers with the column vectors , then with the Euclidean norm ${\displaystyle M}$ ${\displaystyle (n\times n)}$ ${\displaystyle m_{1},\dots ,m_{n}}$ ${\displaystyle \|\cdot \|_{2}}$

${\displaystyle |\det M|\,\leq \,\prod _{i=1}^{n}\|m_{i}\|_{2}.}$

With the QR decomposition of the matrix, the following applies ${\displaystyle M=QR}$ ${\displaystyle M}$

${\displaystyle |\det M|=|\det Q|\cdot |\det R|=|\det R|\leq \|r_{1}\|_{2}\cdot \ldots \cdot \|r_{n}\|_{2},}$

where is. ${\displaystyle \|r_{i}\|_{2}=\|Qr_{i}\|_{2}=\|m_{i}\|_{2}}$

Geometric view

Is a matrix with real entries, then the volume of their row or column vectors spanned dimensional parallelepiped . This volume is maximal for orthogonal rows (or columns) and is consequently at most as large as the volume of the -dimensional cuboid with edges of the lengths . ${\displaystyle M}$ ${\displaystyle (n\times n)}$ ${\displaystyle |\det(M)|}$ ${\displaystyle m_{i}}$ ${\displaystyle n}$ ${\displaystyle \prod _{i=1}^{n}\|m_{i}\|_{2}}$ ${\displaystyle n}$ ${\displaystyle \|m_{i}\|_{2}}$

Attenuated Hadamard Inequality

Let be a commutative ring with pseudo magnitude and a matrix over with the row vectors . Then applies ${\displaystyle (R,|\cdot |)}$ ${\displaystyle M}$ ${\displaystyle (n\times n)}$ ${\displaystyle R}$ ${\displaystyle m_{1},\dots ,m_{n}}$

${\displaystyle |\det M|\,\leq \,\prod _{i=1}^{n}\|m_{i}\|_{1}}$

with the 1- pseudo norm .

Remarks

• Because of this, the classical Hadamard inequality provides the sharper estimate. ${\displaystyle \|x\|_{2}\leq \|x\|_{1}}$
• If a ring is based on the usual absolute value function of complex numbers (example: the whole numbers ), then the more stringent classical Hadamard inequality is always applicable. ${\displaystyle R\subseteq \mathbb {C} }$ ${\displaystyle \mathbb {Z} }$

literature

• Roger A. Horn: Matrix Analysis . Cambridge University Press, 1990, ISBN 978-0-521-38632-6 , pp. 477 ( limited preview in Google Book search).