STATISTICS 512 TECHNIQUES OF MATHEMATICS FOR ...

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• Spectral theory for real, symmetric matrices. First
let M × be any square matrix. For a variable
the determinant |M − I | is a polynomial in
of degree , calledthecharacteristic polynomial.
The equation
|M − I | =0
is the characteristic equation. The Fundamental
Theorem of Algebra states that there are then
(real or complex) roots of this equation. Any
such root is called an eigenvalue of M. If is
an eigenvalue then M − I is singular, so the
columns are dependent:
(M − I ) v =0 (4.1)
for some non-zero vector v (possibly complex),
called the eigenvector corresponding to, or belonging
to, . Thus
Mv = v (4.2)