Page 372 - A Course in Linear Algebra with Applications

356 Chapter Nine: Advanced Topics

In general, if A is any complex n x n matrix, we call a
sequence of vectors X\,..., X T in C n a Jordan string for A if
it satisfies the equations

AX i = cX 1 and AXi = cXi + X;_i

where c is a scalar and 1 < i < r. Thus every n x n Jordan
block determines a Jordan string of length n.
Now suppose there is a basis of C n which consists of
Jordan strings for the matrix A. Group together basis elements
in the same string. Then the linear operator on C n given
by T(X) = AX is represented with respect to this basis of
Jordan strings by a matrix which has Jordan blocks down the
v i 1
- "'fc>^""' -
/Jx 0 °\
0 J 2
N 0
V 0 0 JkJ

Here Jj is a Jordan block, say with Cj on the diagonal. This
is because of the effect produced on the basis elements when
they are multiplied on the left by A.
Our conclusion is that A is similar to the matrix N, which
is called the Jordan normal form of A. Notice that the diagonal
elements Q of N are just the eigenvalues of A. Of course we
still have to establish that a basis consisting of Jordan strings
always exists; only then can we conclude that every matrix
has a Jordan normal form.
Theorem 9.4.5 (Jordan Normal Form)
Every square complex matrix is similar to a matrix in Jordan
normal form.
Proof
Let A be an n x n complex matrix. We have to establish the
n
existence of a basis of C consisting of Jordan strings for A.
This is done by induction on n; if n = 1, any non-zero vector