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 row vectors, then apply with the Euclidean norm

With the QR factorization of the matrix shall namely

Being.

Geometric intuition

Is a real matrix with entries, then the volume of the row vectors of their clamped -dimensional parallelepiped. This volume is at most as large as the volume of -dimensional parallelepiped with edge lengths of the same.

Attenuated Hadamard inequality

Be a commutative ring with pseudo amount and a matrix over the row vectors. Then we have

With the 1- pseudo- norm.

Comments

• The classical Hadamard inequality yields due to the sharper estimate.
• Is a ring with the usual absolute value function of the complex numbers to reason (example: the integers ), the sharper classical Hadamard inequality is always applicable.

• Inequality
• Linear Algebra