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Matakuliah
Tahun
: D0024/Matematika Industri II
: 2008
Matematika
Pertemuan 24
Nilai Eigen dan Vektor Eigen
Let
be a linear transformation represented by a matrix
is a vector
such that
for some scalar
, then
is called the eigenvalue of
corresponding (right) eigenvector
.
Letting
be a
square matrix
. If there
with
(
2
)
with eigenvalue
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, then the corresponding eigenvectors satisfy
with eigenvalue
, then the corresponding eigenvectors satisfy
(
3
)
which is equivalent to the homogeneous system
(
4
)
The last equation can be written compactly as
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where is the identity matrix. As shown in Cramer's rule, a linear system of
equations has nontrivial solutions iff the determinant vanishes, so the solutions
of the last equation are given by
This equation is known as the characteristic equation of
left-hand side is known as the characteristic polynomial.
For example, for a
matrix, the eigenvalues are
(
7
)
which arises as the solutions of the characteristic equation
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, and the
• Kerjakan latihan dalam modul soal
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