Page 155 - Linear Algebra Done Right

Normal Operators on Real Inner-Product Spaces
as you should verify. In other words, to multiply together two block
diagonal matrices (with the same size blocks), just multiply together the
corresponding entries on the diagonal, as with diagonal matrices. 143
A diagonal matrix is a special case of a block diagonal matrix where
each block has size 1-by-1. At the other extreme, every square matrix is Note that if an operator
a block diagonal matrix because we can take the first (and only) block T has a block diagonal
to be the entire matrix. Thus to say that an operator has a block di- matrix with respect to
agonal matrix with respect to some basis tells us nothing unless we some basis, then the
know something about the size of the blocks. The smaller the blocks, entry in any 1-by-1
the nicer the operator (in the vague sense that the matrix then contains block on the diagonal
more 0’s). The nicest situation is to have an orthonormal basis that of this matrix must be
gives a diagonal matrix. We have shown that this happens on a com- an eigenvalue of T.
plex inner-product space precisely for the normal operators (see 7.9)
and on a real inner-product space precisely for the self-adjoint opera-
tors (see 7.13).
Our next result states that each normal operator on a real inner-
product space comes close to having a diagonal matrix—specifically,
we get a block diagonal matrix with respect to some orthonormal basis,
with each block having size at most 2-by-2. We cannot expect to do bet-
ter than that because on a real inner-product space there exist normal
operators that do not have a diagonal matrix with respect to any basis.
2
For example, the operator T ∈L(R ) defined by T(x, y) = (−y, x) is
normal (as you should verify) but has no eigenvalues; thus this partic-
ular T does not have even an upper-triangular matrix with respect to
2
any basis of R .
Note that the matrix in 7.22 is the type of matrix promised by the
theorem below. In particular, each block of 7.22 (see 7.23) has size
at most 2-by-2 and each of the 2-by-2 blocks has the required form
(upper left entry equals lower right entry, lower left entry is positive,
and upper right entry equals the negative of lower left entry).

7.25 Theorem: Suppose that V is a real inner-product space and
T ∈L(V). Then T is normal if and only if there is an orthonormal
basis of V with respect to which T has a block diagonal matrix where
each block is a 1-by-1 matrix or a 2-by-2 matrix of the form

a −b
7.26 ,
b a
with b> 0.