MTH 261 – Mr. Simonds’ class – Determinants and Eigenvalues/Eigenvectors
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Example
Show that the matrix
211
211
11 2
A
⎡⎤
⎢⎥
=−−
⎢⎥
⎢⎥
−
⎣⎦
is deficient. You may use your calculator to find the
eigenvalues.
Terminology and theorems regarding eigenvalues and eigenvectors
Let
A
be an
nn×
matrix.
•
A
is invertible if and only if 0 is not an eigenvalue of
A
.
• For any given eigenvalue of
A
,
i
λ
,
(
)
()
1dim mult
i
i
V
λ
λ
≤≤
where
i
V
λ
is the
eigenspace associated with
i
λ
and
(
)
mult
i
λ
is the multiplicity of
i
λ
in the
characteristic equation of
A
.
• An eigenbasis for
A
is a bases for
n
\ consisting of eigenvectors of
A
.
A
has
an eigenbasis if and only if
(
)
(
)
dim mult
i
i
V
λ
λ
= for each eigenvalue
i
λ
. A
matrix that has any eigenvalue where
(
)
()
dim mult
i
i
V
λ
λ
< is said to be deficient.
