Page 124 - Linear Algebra Done Right

Orthogonal Projections and Minimization Problems
for each j (see 6.21), we conclude that span(e 1 ,...,e j ) is invariant un-
der T for each j = 1,...,n. Thus, by 5.12, T has an upper-triangular
matrix with respect to the orthonormal basis (e 1 ,...,e n ). 111
The next result is an important application of the corollary above.
6.28 Corollary: Suppose V is a complex vector space and T ∈L(V). This result is
Then T has an upper-triangular matrix with respect to some orthonor- sometimes called
mal basis of V. Schur’s theorem. The
German mathematician
Proof: This follows immediately from 5.13 and 6.27. Issai Schur published
the first proof of this
result in 1909.
Orthogonal Projections and
Minimization Problems

If U is a subset of V, then the orthogonal complement of U, de-
noted U , is the set of all vectors in V that are orthogonal to every
⊥
vector in U:

U ⊥ ={v ∈ V : v, u = 0 for all u ∈ U}.

You should verify that U ⊥ is always a subspace of V, that V ⊥ ={0},
⊥
and that {0} = V. Also note that if U 1 ⊂ U 2 , then U 1 ⊥ ⊃ U .
⊥
2
Recall that if U 1 , U 2 are subspaces of V, then V is the direct sum of
U 1 and U 2 (written V = U 1 ⊕ U 2 ) if each element of V can be written in
exactly one way as a vector in U 1 plus a vector in U 2 . The next theorem
shows that every subspace of an inner-product space leads to a natural
direct sum decomposition of the whole space.

6.29 Theorem: If U is a subspace of V, then

⊥
V = U ⊕ U .

Proof: Suppose that U is a subspace of V. First we will show that
6.30 V = U + U .
⊥

To do this, suppose v ∈ V. Let (e 1 ,...,e m ) be an orthonormal basis
of U. Obviously