Algebraic and Geometric
Multiplicities of Eigenvalues
Let: T : V → V be a linear operator on a finitedimensional K-vector space V.
Definition: The algebraic multiplicity of λ (as an
eigenvalue of T ) is its multiplicity as a root of chT ;
we denote it by
mλ(T ) = multλ(chT )
= max{m : chT (x) = (x−λ)mh(x),
for some h(x) ∈ K[x]}.
The geometric multiplicity of λ (as an eigenvalue
of T ) is
gλ(T ) := dim Eλ(T ) = dim(V ) − rank(T − λIV ).
The algebraic and geometric multiplicities of λ
with respect to A ∈ Mn(K) are defined by
mλ(A) = mλ(TA) and gλ(A) = gλ(TA).
Theorem 17: If B is any basis of V , then
mλ([T ]B ) = mλ(T ) and gλ([T ]B ) = gλ(T ).
2
Corollary: If A ∼ B are two similar matrices, i.e. if
A = P BP −1 for some invertible matrix P , then
mλ(A) = mλ(B) and gλ(A) = gλ(B).
Remark: By the Corollary of Theorem 16 we have
that
X
(1)
gλ(T ) ≤ dim(V ).
λ∈spec(T )
Analogously we have the inequality
X
(2)
mλ(T ) ≤ dim(V ).
λ∈spec(T )
Q
∗
Indeed, write chT (x) = λ∈spec(T )(x − λ)mλ(T ).
Then we have that
chT (x) = ch∗T (x)h(x),
where h(x) ∈ K[x] is a (monic) polynomial with no
roots in K. From (3) the inequality (2) follows by
taking degrees. Moreover, we thus see that
(3)
Equality holds in (2) ⇔ ch∗T (x) = chT (x)
⇔ chT (x) splits completely
over K.
3
Theorem 18: We have that
(4)
gλ(T ) ≤ mλ(T ), for all λ ∈ spec(T ).
Theorem 19 (Diagonalization Theorem): The
following conditions are equivalent:
(a) T is diagonable;
(b) equality holds in (1);
(c) mλ(T ) = gλ(T ), for all λ ∈ spec(T ) and chT (x)
splits completely over K.
Remark: The splitting condition is missing from the
statement of Theorem 4.27 of the text. It seems
that the book assumes here that K = C (so all
diagonalization takes place over C), but this is not
stated explicitly.
h
i
Example: Let A = 01 −10 .
Since chA(λ) = λ2 + 1 has no real roots, A has no
real eigenvalues and A is not diagonable over R.
However, chA(λ) = (λ − i)(λ + i) in C[λ], so A can
be diagonalized over C. Explicitly:
h
i h
ih
ih
i−1
A = 01 −10 = −i1 1i i0 −i0 −i1 1i
.