Linear Algebra, 2020a

Section IV. Jordan Form 441
The polynomial of the matrix represents the polynomial of the map: if T =
Rep B,B (t) then f(T) =Rep B,B (f(t)). This is because T j = Rep B,B (t j ), and
cT = Rep B,B (ct), and T 1 + T 2 = Rep B,B (t 1 + t 2 ).
1.2 Remark We shall write the matrix polynomial slightly differently than the
map polynomial. For instance, if f(x) =x − 3 then we shall write the identity
matrix, as in f(T) =T − 3I, but not write the identity map, as in f(t) =t − 3.
1.3 Example Rotation of plane vectors π/3 radians counterclockwise is represented
with respect to the standard basis by
(
) (
cos(π/3) −sin(π/3) 1/2 − √ )
3/2
T =
= √
sin(π/3) cos(π/3) 3/2 1/2
and verifying that T 2 − T + I = Z is routine. (Geometrically, T 2 rotates by 2π/3.
On the standard basis vector ⃗e 1 , the action of T 2 − T is to give the difference
between the unit vector with angle 2π/3 and the unit vector with angle π/3,
which is ( )
−1
0 .On⃗e2 it gives ( 0
−1)
.SoT 2 − T =−I.)
The space M 2×2 has dimension four so we know that for any 2×2 matrix T
there is a fourth degree polynomial f such that f(T) =Z. But in that example
we exhibited a degree two polynomial that works. So while degree n 2 always
suffices, in some cases a smaller-degree polynomial is enough.
1.4 Definition The minimal polynomial m(x) of a transformation t or a square
matrix T is the non-zero polynomial of least degree and with leading coefficient 1
such that m(t) is the zero map or m(T) is the zero matrix.
A minimal polynomial cannot be a constant polynomial because of the restriction
on the leading coefficient. So a minimal polynomial must have degree at least
one. The zero matrix has minimal polynomial p(x) =x while the identity matrix
has minimal polynomial ˆp(x) =x − 1.
1.5 Lemma Any transformation or square matrix has a unique minimal polynomial.
Proof First we prove existence. By the earlier observation that degree n 2
suffices, there is at least one nonzero polynomial p(x) =c k x k + ···+ c 0 that
takes the map or matrix to zero. From among all such polynomials take one
with the smallest degree. Divide this polynomial by its leading coefficient c k to
get a leading 1. Hence any map or matrix has at least one minimal polynomial.
Now for uniqueness. Suppose that m(x) and ˆm(x) both take the map or
matrix to zero, are both of minimal degree and are thus of equal degree, and
both have a leading 1. Consider the difference, m(x)− ˆm(x). If it is not the zero