Page 123 - Linear Algebra Done Right

110
the Gram-Schmidt procedure (6.20) to (e 1 ,...,e m ,v 1 ,...,v n ), produc-
ing an orthonormal list
(e 1 ,...,e m ,f 1 ,...,f n );
6.26 Chapter 6. Inner-Product Spaces
here the Gram-Schmidt procedure leaves the first m vectors unchanged
because they are already orthonormal. Clearly 6.26 is an orthonormal
basis of V because it is linearly independent (by 6.16) and its span
equals V. Hence we have our extension of (e 1 ,...,e m ) to an orthonor-
mal basis of V.

Recall that a matrix is called upper triangular if all entries below the
diagonal equal 0. In other words, an upper-triangular matrix looks like
this:
 
∗ ∗
 . 
 . .  .
 
0 ∗
In the last chapter we showed that if V is a complex vector space, then
for each operator on V there is a basis with respect to which the matrix
of the operator is upper triangular (see 5.13). Now that we are dealing
with inner-product spaces, we would like to know when there exists an
orthonormal basis with respect to which we have an upper-triangular
matrix. The next corollary shows that the existence of any basis with
respect to which T has an upper-triangular matrix implies the existence
of an orthonormal basis with this property. This result is true on both
real and complex vector spaces (though on a real vector space, the hy-
pothesis holds only for some operators).

6.27 Corollary: Suppose T ∈L(V).If T has an upper-triangular
matrix with respect to some basis of V, then T has an upper-triangular
matrix with respect to some orthonormal basis of V.

Proof: Suppose T has an upper-triangular matrix with respect to
some basis (v 1 ,...,v n ) of V. Thus span(v 1 ,...,v j ) is invariant under
T for each j = 1,...,n (see 5.12).
Apply the Gram-Schmidt procedure to (v 1 ,...,v n ), producing an
orthonormal basis (e 1 ,...,e n ) of V. Because

span(e 1 ,...,e j ) = span(v 1 ,...,v j )