36. Hadamard's determinant estimate

In Section 25, it has been shown that, if a₁, ··· , a_n are linearly independent (n,1)-vectors, numbers b₁₁, ··· b_nn can be chosen so that

c₁ = b₁₁a₁,
c₂ = b₂₁c₁ + b₂₂a₂,
c₃ = b₃₁c₁ + b₃₂a₂ + b₃₃a₃,
················································
c_n = b_n1c₁ + b_n2a₂ + ··· + b_nna_n

form a normalized orthogonal system. Due to the linear independence of the c₁,···, c_n, all the b_ii 0, so that you can solve the system of equation for the a_i and obtain

a₁ = p₁₁c₁,
a₂ = p₁₂c₁ + p₂₂c₂,
···························································· (36.1)
a_n = p_1nc₁ + p_2nc₂ + ··· + p_nnc_n

These equations can be represented by a single matrix equation. Treat the a₁,···,a_n as the columns of a matrix A, the c₁,···, c_n as the columns of an unitary matrix C and set

(36.1) can now be written in the form A = CP, whence

Since is again unitary, you find: Every regular matrix A can be converted into a triangular matrix by multiplication from the left by a unitary matrix. Taking on both sides the absolute values of the determinants and taking into consideration that |||| = 1, you find

Let now all the vectors a₁,···,a_n be normalized:

Since the c_i, are orthogonal to each other and normalized, (36.1) yields

Hence |p_ii| 1. Since ||P|| = |p₁₁···p_nn|| 1, (36.3) yields

||A|| 1 (36.5)

The equality sign only occurs when all |p_ii| = 1. However, by (36.4), all p_ik with
i k vanish, so that P becomes a diagonal matrix. Moreover, this means that

a_i = p_iic_i with |p_ii| = 1 (i = 1, ··· , n),

so that the a_i themselves are mutually orthogonal. Hence follows:

If A is a matrix, the columns of which form a system of orthogonalized vectors, then always ||A|| 1 and the equality sign only occurs if A is a unitary matrix.

The inequality ||A|| 1 also applies trivially if the columns of A are linearly dependent, since then |A| = 0.

If we now multiply the columns of A in sequence by the positive numbers a₁,···,a_n, the determinant |A| is multiplied by the product a₁···a_n and you have for the norms of the columns

The inequality (36.5) now yields:

Provided (36.6) applies, there applies Hadamard's estimate

||A|| a₁···a_n.

The equality sign only arises when the columns of A are mutually orthogonal. A corresponding estimate arises with regard to the norms of the rows. *

* As Analytic Geometry shows, this statement is the n-dimensional analogue to the rules that of all parallelograms with given side lengths a₁, a₂ the rectangle has the largest area and that of all parallelepipeds with given edge lengths a₁, a₂, a₃ the cube has the largest volume.

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