18. Contragredient and orthogonal matrices

If U and V are the matrices of the types (n,p) and (p,q) considered in Section 14, the transposed matrices U' and V' are of the types (p,n) and (q,p), respectively, whence V' is linked to U' and

where the sums must be taken from j = 1, ··· , p. The comparison with the product UV computed in Section 14 yields

(UV)' = V'U'.

By induction, you find for the transposition of a product of more than two factors

(U₁U₂ ··· U_k-1U_k)' = U'_kU'_k-1 ··· U'₂U'₁. (18.1)

Obviously, for every matrix, (U')' = U.

Transition to the transposed matrix yields A'(A^-1)'= E, whence

(A')^-1 = (A^-1)', (18.2)

an equation which also follows from (15.4). The inner product

x₁y₁ + x₂y₂ + ··· + x_ny_n

of the two sets of variables x₁, ··· , x_n and y₁, ··· , y_n is written conveniently in matrix notation. In fact, if x and y denote the column vectors x₁, ··· , x_n and y₁, ··· , y_n, then

* No distinction is made between a (1,1) matrix, consisting of one element, and the element itself.

The product y'x has the same value, as can be computed or concluded from the fact that x'y as (1,1) matrix agrees with its transpose, since this observtion yields with (18.1)

x'y = (x'y)' = y'x.

Since inner products occur frequently as scalar products of vectors, importance is attached to the question regarding the behaviour of the inner product of two sequences of variables during linear substitutions. Let u₁,···,u_n and v₁,···,v_n denote the columns of the variables and let them be related to x and y through the regular, linear substitutions

u = Ax, v = By. (18.3)

According to (18.1), the inner product is

u'v = x'A'By. (18.4)

Frequently, one demands that linear substitutions do not change the inner product. The equation

u₁v₁ + ··· + u_nv_n = x₁y₁ + ··· + x_ny_n (18.5)

or u'v = x'y means, due to (18.4), the same as A'B = E or B = (A')^-1, which is called the matrix, contragredient to A. Due to (18.2), the matrix, contragredient to (A')^-1is again A. We can now state:

Theorem 6

Equation (18.5) applies if and only if the linear substitutions (18.3), which convert the variables x into u and the variables y into v, are contragredient with respect to each other.

In the case that all variables are subject to the same linear substitution and that in the process the inner products are to remain invariant, then there must be in (18.3) A=B and, besides, B = (A')^-1, that is, A = (A')^-1. This property of the matrix can also be expressed by the equation

AA' = E. (18.6)

A is then said to be an orthogonal matrix.

Theorem 7

If u = Ax and v = Ay, then and only then u'v = x'y, if A is orthogonal,

whence especially:

If the variables u and v are interrelated by the regular, linear substitution u = Ax, then and only then

u₁² + ··· + u_n² = x₁² + ··· + x_n²,

if A is orthogonal.

The characteristic property (18.6) of an orthogonal matrix can also be formulated as follows: The inner product of each row of A with itself has the value 1; the inner product of any two different rows of A vanishes. The relation A'A = E means the same as (18.6). whence you can also characterize an orthogonal matrix A by the property: The inner product of any column of A with itself is 1, the inner product of two different columns of A vanishes.

Finally, we will prove that the product of two orthogonal matrices is again orthogonal. In fact, if C = AB, where A and B are orthogonal, that is, AA' = E and BB' = E, then

CC' = ABB'A' = AEA' = AA' = E,

whence you draw readily the conclusion that the totality of all orthogonal (n,n)-matrices forms a group (cf. Section 16).

Exercises

14. Let D be a diagonal matrix with all different diagonal elements. Find all matrices which can be interchanged with D.
15. Denote by A^* the matrix, contragredient to A. Prove that (AB)^* = A^*B^*.
16. The determinant of an orthogonal matrix has the value +1 or -1.
17. All two-rows, real, orthogonal matrices have the form

depending on whether the determinant has the value +1 or -1.

Answers

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