Page 205 - Linear Algebra Done Right

Block Upper-Triangular Matrices
We have
Ax =M(T)M(v) =M(Tv) =M(λv) = λM(v) = λx,
where the second equality comes from 3.14. The equation above shows 195
that λ is an eigenvalue of A, as desired.
To prove the implication in the other direction, now suppose λ is an
eigenvalue of A. Let x be a nonzero n-by-1 matrix such that Ax = λx.
We can write x in the form 9.3 for some scalars a 1 ,...,a n ∈ F. Define
v ∈ V by 9.2. Then
M(Tv) =M(T)M(v) = Ax = λx =M(λv).

where the first equality comes from 3.14. The equation above implies
that Tv = λv, and thus λ is an eigenvalue of T, completing the proof.

Because every square matrix is the matrix of some operator, the
proposition above allows us to translate results about eigenvalues of
operators into the language of eigenvalues of square matrices. For
example, every square matrix of complex numbers has an eigenvalue
(from 5.10). As another example, every n-by-n matrix has at most n
distinct eigenvalues (from 5.9).

Block Upper-Triangular Matrices

Earlier we proved that each operator on a complex vector space has
an upper-triangular matrix with respect to some basis (see 5.13). In
this section we will see that we can almost do as well on real vector
spaces.
In the last two chapters we used block diagonal matrices, which
extend the notion of diagonal matrices. Now we will need to use the
corresponding extension of upper-triangular matrices. A block upper-
triangular matrix is a square matrix of the form As usual, we use an
asterisk to denote
 
A 1 ∗
 .  entries of the matrix
 . .  , that play no important
 
0 A m role in the topics under
consideration.
where A 1 ,...,A m are square matrices lying along the diagonal, all en-
tries below A 1 ,...,A m equal 0, and the ∗ denotes arbitrary entries. For
example, the matrix