41. Cayley-Hamilton relations

Let

j (x) = |xE - A| = xⁿ + a_n-1x^n-1+ ··· + a₀

be the characteristic polynomial of the (n,n)-matrix A. According to (15.4) for the formation of the inverse matrix, the elements of (xE - A)^-1 have the form: They are the quotients of polynomials in x; the common denominator of all the quotients is j (x); the numerator are the minor determinants of order n-1 of xE-A, whence the numerators have at the most the degree n - 1 and you can write

(xE - A)^-1 = B(x)/j (x), (41.1)

where the elements of the matrix B(x) are polynomials of at the most degree n-1 in x. Hence the matrix B(x) can be given the form

B(x) = B₀xⁿ + B₁x + ··· + B_n-1x^n-1

with constant (independent of x matrices B₀, ··· , B_n-1). (41.1) yields

j (x)E = (xE - A)B(x) = (xE - A)(B₀xⁿ + B₁x + ··· + B_n-1x^n-1),
(xⁿ + a_n-1x^n-1+···+a₀)E = x^n-1B_n-1 + x^n-1(B_n-2 - AB_n-1) + (B_n-3 - AB_n-2)+···+x(B₀ - AB₁)- AB₀.

Comparing the matrices with equal powers of x on both sides, you find

E = B_n-1,
a_n-1E = B_n-2 - AB_n-1,
a_n-2E = B_n-3 - AB_n-2,
··········································
aE = B₀ - AB_1,
a₀E = - AB₀.

Multiplying these equations in turn by Aⁿ, A^n-1, ··· , A, E and adding them, you find

Aⁿ + a_n-1A^n-1 + a_n-1A^n-2 + ··· + a₁A + a₀E = 0 (41.2)
_{or
j (A) = 0,}

_{the Cayley-Hamilton relation.}

If |A| 0, then the coefficient a₀ in j (x) does no vanish and

a₀ = (-1)ⁿ|A|,

whence, on multiplication by A^-1, you find for A^-1 the polynomial in A

A^-1 = (A^n-1 + a_n-1A^n-2 + a_n-2A^n-3 + ··· + a₁E).

Exercises

27. Prove that

where s_m is the m-th sum

s_m = x₁^m + x₂^m + ··· + x_n^m (m = 0, 1, ··· ; s = n).

The product is called the discriminant of x₁, x₂, ··· + x_n.

28. With A = (a_ik) and a = max(|a₁₁|, |a₁₂|, ··· , |a_nn|), every characteristic root a of A satisfies the inequality |a| 1. Prove this estimate and show that the bounds of na cannot be reduced (Tip: Consider one (n,n)-matrix F, all elements of which are 1.)

29. Let a be a characteristic root of A. If A is skew-symmetric, that is, A' = - A, then also -a is a characteristic root. If A is orthogonal, then a^-1 is also a characteristic root.

30. All characteristic roots of a unitary matrix have the absolute value 1.

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