Linear Algebra II (pdf, 500 kB)

2
1. Review of Eigenvalues, Eigenvectors and Characteristic
Polynomial
Recall the topics we finished Linear Algebra I with. We were discussing eigenvalues
and eigenvectors of endomorphisms and square matrices, and the question when
they are diagonalizable. For your convenience, I will repeat here the most relevant
definitions and results.
Let V be a finite-dimensional F -vector space, dim V = n, and let f : V → V be
an endomorphism. Then for λ ∈ F , the λ-eigenspace of f was defined to be
Eλ(f) = {v ∈ V : f(v) = λv} = ker(f − λ idV ) .
λ is an eigenvalue of f if Eλ(f) = {0}, i.e., if there is 0 = v ∈ V such that
f(v) = λv. Such a vector v is called an eigenvector of f for the eigenvalue λ.
The eigenvalues are exactly the roots (in F ) of the characteristic polynomial of f,
Pf(x) = det(x idV −f) ,
which is a monic polynomial of degree n with coefficients in F .
The geometric multiplicity of λ as an eigenvalue of f is defined to be the dimension
of the λ-eigenspace, whereas the algebraic multiplicity of λ as an eigenvalue of f
is defined to be its multiplicity as a root of the characteristic polynomial.
The endomorphism f is said to be diagonalizable if there exists a basis of V
consisting of eigenvectors of f. The matrix representing f relative to this basis is
then a diagonal matrix, with the various eigenvalues appearing on the diagonal.
Since n × n matrices can be identified with endomorphisms F n → F n , all notions
and results makes sense for square matrices, too. A matrix A ∈ Mat(n, F ) is
diagonalizable if and only if it is similar to a diagonal matrix, i.e., if there is an
invertible matrix P ∈ Mat(n, F ) such that P −1 AP is diagonal.
It is an important fact that the geometric multiplicity of an eigenvalue cannot
exceed its algebraic multiplicity. An endomorphism or square matrix is diagonalizable
if and only if the sum of the geometric multiplicities of all eigenvalues equals
the dimension of the space. This in turn is equivalent to the two conditions (a)
the characteristic polynomial is a product of linear factors, and (b) for each eigenvalue,
algebraic and geometric multiplicities agree. For example, both conditions
are satisfied if Pf is the product of n distinct monic linear factors.
2. The Cayley-Hamilton Theorem and the Minimal Polynomial
Let A ∈ Mat(n, F ). We know that Mat(n, F ) is an F -vector space of dimension n2 .
Therefore, the elements I, A, A2 , . . . , An2 cannot be linearly independent (because
their number exceeds the dimension). If we define p(A) in the obvious way for p
a polynomial with coefficients in F , then we can deduce that there is a (non-zero)
polynomial p of degree at most n2 such that p(A) = 0 (0 here is the zero matrix).
In fact, much more is true.
Consider a diagonal matrix D = diag(λ1, λ2, . . . , λn). (This notation is supposed
to mean that λj is the (j, j) entry of D; the off-diagonal entries are zero, of course.)
Its characteristic polynomial is
PD(x) = (x − λ1)(x − λ2) · · · (x − λn) .
Since the diagonal entries are roots of PD, we also have PD(D) = 0. More generally,
consider a diagonalizable matrix A. Then there is an invertible matrix Q such