Linear Algebra, 2020a

440 Chapter Five. Similarity
IV
Jordan Form
This section uses material from three optional subsections: Combining
Subspaces, Determinants Exist, and Laplace’s Expansion.
We began this chapter by recalling that every linear map h: V → W can be
represented with respect to some bases B ⊂ V and D ⊂ W by a partial identity
matrix. Restated, the partial identity form is a canonical form for matrix
equivalence. This chapter considers the case where the codomain equals the
domain so we naturally ask what is possible when the two bases are equal, when
we have Rep B,B (t). In short, we want a canonical form for matrix similarity.
We noted that in the B, B case a partial identity matrix is not always possible.
We therefore extended the matrix forms of interest to the natural generalization,
diagonal matrices, and showed that a transformation or square matrix can be
diagonalized if its eigenvalues are distinct. But we also gave an example of a
square matrix that cannot be diagonalized because it is nilpotent, and thus
diagonal form won’t suffice as the canonical form for matrix similarity.
The prior section developed that example to get a canonical form for nilpotent
matrices, subdiagonal ones.
This section finishes our program by showing that for any linear transformation
there is a basis B such that the matrix representation Rep B,B (t) is the sum
of a diagonal matrix and a nilpotent matrix. This is Jordan canonical form.
IV.1
Polynomials of Maps and Matrices
Recall that the set of square matrices M n×n is a vector space under entry-byentry
addition and scalar multiplication, and that this space has dimension n 2 .
Thus for any n×n matrix T the n 2 + 1-member set {I, T, T 2 ,...,T n2 } is linearly
dependent and so there are scalars c 0 ,...,c n 2, not all zero, such that
c n 2T n2 + ···+ c 1 T + c 0 I
is the zero matrix. Therefore every transformation has a sort of generalized
nilpotency — the powers of a square matrix cannot climb forever without a kind
of repeat.
1.1 Definition Let t be a linear transformation of a vector space V. Where
f(x) =c n x n + ···+ c 1 x + c 0 is a polynomial, f(t) is the transformation c n t n +
···+ c 1 t + c 0 (id) on V. In the same way, if T is a square matrix then f(T) is
the matrix c n T n + ···+ c 1 T + c 0 I.