Linear Algebra, Theory And Applications, 2012a

310 SELF ADJOINT OPERATORS
It follows easily that ˜F is also a commuting family of diagonalizable matrices. By
Lemma 13.1.7 every B ∈ ˜F is of the form given in (13.2) because each of these commutes
with D described above as S −1 AS and so by block multiplication, the diagonal blocks B i
corresponding to different B ∈ ˜F commute.
By Corollary 13.1.4 each of these blocks is diagonalizable. This is because B is known to
be so. Therefore, by induction, since all the blocks are no larger than n−1×n−1 thanks to
the assumption that A has more than one eigenvalue, there exist invertible n i × n i matrices,
T i such that T −1
i B i T i is a diagonal matrix whenever B i is one of the matrices making up
the block diagonal of any B ∈F. It follows that for T defined by
⎛
⎞
T 1 0 ··· 0
.
T ≡
0 T .. . 2 ⎜
⎝
.
. .. . ⎟ .. 0 ⎠ ,
0 ··· 0 T r
then T −1 BT = a diagonal matrix for every B ∈ ˜F including D. Consider ST. It follows
that for all C ∈F,
something in
{ }} {
˜F
T −1 S −1 CS T =(ST) −1 C (ST) = a diagonal matrix.
Theorem 13.1.9 Let F denote a family of matrices which are diagonalizable. Then F is
simultaneously diagonalizable if and only if F is a commuting family.
Proof: If F is a commuting family, it follows from Lemma 13.1.8 that it is simultaneously
diagonalizable. If it is simultaneously diagonalizable, then it follows from Lemma 13.1.6 that
it is a commuting family.
13.2 Schur’s Theorem
Recall that for a linear transformation, L ∈L(V,V )forV a finite dimensional inner product
space, it could be represented in the form
L = ∑ ij
l ij v i ⊗ v j
where {v 1 , ··· , v n } is an orthonormal basis. Of course different bases will yield different
matrices, (l ij ) . Schur’s theorem gives the existence of a basis in an inner product space such
that (l ij ) is particularly simple.
Definition 13.2.1 Let L ∈L(V,V ) where V is vector space. Then a subspace U of V is L
invariant if L (U) ⊆ U.
In what follows, F will be the field of scalars, usually C but maybe something else.
Theorem 13.2.2 Let L ∈L(H, H) for H a finite dimensional inner product space such
that the restriction of L ∗ to every L invariant subspace has its eigenvalues in F. Then there
exist constants, c ij for i ≤ j and an orthonormal basis, {w i } n i=1
such that
L =
n∑
j=1 i=1
The constants, c ii are the eigenvalues of L.
j∑
c ij w i ⊗ w j