Page 232 - Linear Algebra Done Right

Determinant of an Operator
has multiplicity 1) and (−4, 13) is the only eigenpair of T (it has multi-
plicity 1). Computing the product of the eigenvalues times the product
of the second coordinates of the eigenpairs, we have det T = (1)(13); 223
in other words, det T = 13.
The reason that the operators in the two previous examples have the
same determinant will become clear after we find a formula (valid on
both complex and real vector spaces) for computing the determinant
of an operator from its matrix.
In this section, we will prove some simple but important properties
of determinants. In the next section, we will discover how to calculate
det T from the matrix of T (with respect to an arbitrary basis). We begin
with a crucial result that has an easy proof with our approach.

10.14 Proposition: An operator is invertible if and only if its deter-
minant is nonzero.

Proof: First suppose V is a complex vector space and T ∈L(V).
The operator T is invertible if and only if 0 is not an eigenvalue of T.
Clearly this happens if and only if the product of the eigenvalues of T
is not 0. Thus T is invertible if and only if det T = 0, as desired.
Now suppose V is a real vector space and T ∈L(V). Again, T is
invertible if and only if 0 is not an eigenvalue of T. Using the notation
of 10.7, we have

10.15 det T = λ 1 ...λ m β 1 ...β M ,
where the λ’s are the eigenvalues of T and the β’s are the second coor-
dinates of the eigenpairs of T, each repeated according to multiplicity.
2
For each eigenpair (α j ,β j ), we have α j < 4β j . In particular, each β j
is positive. This implies (see 10.15) that λ 1 ...λ m = 0 if and only if
det T = 0. Thus T is invertible if and only if det T = 0, as desired.

If T ∈L(V) and λ, z ∈ F, then λ is an eigenvalue of T if and only if
z − λ is an eigenvalue of zI − T. This follows from

−(T − λI) = (zI − T) − (z − λ)I.

Raising both sides of this equation to the dim V power and then taking
null spaces of both sides shows that the multiplicity of λ as an eigen-
value of T equals the multiplicity of z − λ as an eigenvalue of zI − T.