Page 363 - A Course in Linear Algebra with Applications

9.4: Jordan Normal Form 347

8. Find the canonical form of the skew-symmetric matrix

T
and also find an invertible matrix S such that S AS equals
the canonical form.
9. (a) If A is a square matrix and S is an invertible matrix,
T
prove that A and S AS have the same rank.
(b) Deduce that the rank of a skew-symmetric matrix
equals twice the number of 2 x 2 blocks in the canonical form
of the matrix. Conclude that the canonical form is unique.

10. Call a skew-symmetric bilinear form / on a vector space
V non-isotropic if for every non-zero vector v there is an-
other vector w in V such that /(v, w) ^ 0. Prove that a
finite-dimensional real or complex vector space which has a
non-isotropic skew-symmetric bilinear form must have even
dimension.

9.4 Minimum Polynomials and Jordan Normal Form

The aim of this section is to introduce the reader to one
of the most famous results in linear algebra, the existence of
what is known as Jordan normal form of a matrix. This is a
canonical form which applies to any square complex matrix.
The existence of Jordan normal form is often presented as the
climax of a series of difficult theorems; however the simpli-
fied approach adopted here depends on only elementary facts
about vector spaces. We begin by introducing the important
concept of the minimum polynomial of a linear operator or
matrix.