Proof
The result is trivial if the matrix N is singular, so assume the columns of N are linearly independent. By dividing each column by its length, it can be seen that the result is equivalent to the special case where each column has length 1, in other words if ei are unit vectors and M is the matrix having the ei as columns then
and equality is achieved if and only if the vectors are an orthogonal set, that is when the matrix is unitary. The general result now follows:
For the positive definite case, let P =M*M and let the eigenvalues of P be λ1, λ2, … λn. By assumption, each entry in the diagonal of P is 1, so the trace of P is n. Applying the inequality of arithmetic and geometric means,
so
If there is equality then each of the λi's must all be equal and their sum is n, so they must all be 1. The matrix P is Hermitian, therefore diagonalizable, so it is the identity matrix—in other words the columns of M are an orthonormal set and the columns of N are an orthogonal set.
Many other proofs can be found in the literature.
Read more about this topic: Hadamard's Inequality
Famous quotes containing the word proof:
“War is a beastly business, it is true, but one proof we are human is our ability to learn, even from it, how better to exist.”
—M.F.K. Fisher (1908–1992)
“There are some persons in this world, who, unable to give better proof of being wise, take a strange delight in showing what they think they have sagaciously read in mankind by uncharitable suspicions of them.”
—Herman Melville (1819–1891)
“To cease to admire is a proof of deterioration.”
—Charles Horton Cooley (1864–1929)