Page 323 - DMTH502_LINEAR_ALGEBRA

Unit 29: Spectral Theory and Properties of Normal Operators

p = (x – c) (x – c ) Notes
j
where c is a non-real complex number.
Now let f = p/p . Then, since f , . . ., f are relatively prime, there exist polynomials g with
j j 1 k j
coefficients in the scalar field such that

1 = f g . … (6)
j j
j
We briefly indicate how such g may be constructed. If p = x – c , then f (c )  0, and for g we take
j j j j j j
the scalar polynomial 1/f (c ). When every p is of this form, the f g are the familiar Lagrange
j j j j j
polynomials associated with c , . . ., c , and (6) is clearly valid. Suppose some p = (x – c)(x – c )
1 k j
with c a non-real complex number. Then V is a real inner product space, and we take
x c x c
g =
j
s s
where s = (c – c ) f (c). Then
j
(s s )x (cs cs )
g =
j
ss
so that g is a polynomial with real coefficients. If p has degree n, then
j
1 – f g j
j
j
is a polynomial with real coefficients of degree at almost n – 1; moreover, it vanishes at each of
the n (complex) roots of p, and hence is identically 0.
Now let be an arbitrary vector in V. Then by (16)

= f ( )g ( )
T
T
j j
j
and since p (T) f (T) = 0, it follows that f (T) g (T) is in W for every j. By Lemma 4, W is orthogonal
j j j j j j
to W whenever i j. Therefore, V is the orthogonal direct sum of W , . . ., W . If is any vector in
j 1 k
W , then
j
p (T) T = Tp (T) = 0;
j j
thus W is invariant under T. Let T be the restriction of T to W . Then p (T ) = 0, so that p is divisible
j j j j j j
by the minimal polynomial for T . Since p is irreducible over the scalar field, it follows that p is
j j j
the minimal polynomial for T .
j
Next, let e = f g and E = e (T). Then for every vector in V, E is in W , and
j j j j j j j
= E j
j
Thus – E = E since W is orthogonal to W when j i, this implies that – E is in W . It
j j j j j i
j i
now follows from Theorem 4 of unit 24 that E is the orthogonal projection of V on W .
i i
Definition: We call the subspaces W (1 j k) the primary components of V under T.
j
Corollary: Let T be a normal operator on a finite-dimensional inner product space V and W ,
1
…, W the primary components of V under T. Suppose W is a subspace of V which is invariant
k
under T.

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