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(For the ranges below –2 and above 2, the hyperbolic cosine function will be
more appropriate and similar steps to the ones that we will follow can be
repeated.)
Having found the eigenvalues, which can now be expressed in the simple
form:

± θ j
λ = e (8.98)
±
let us proceed to find the matrix V, defined as:

M = VDV −1 or MV = VD (8.99)

and where D is the diagonal matrix of the eigenvalues. By direct substitution,
in the matrix equation defining V, Eq. (8.99), the following relations can be
directly obtained:

V λ − d
11 = + (8.100)
V c
21

and

V λ − d
12 = − (8.101)
V c
22
If we choose for convenience V = V = c (which is always possible because
11
22
each eigenvector can have the value of one of its components arbitrary cho-
sen with the other components expressed as functions of it), the matrix V can
be written as:

e jθ − d e − jθ −  d
V =   (8.102)
 c c 

and the matrix M can be then written as:

e jθ − d e − jθ −  e jθ 0   c d − e − jθ 
d
    − jθ    jθ −  
M = c c   0 e  −c e d (8.103)
j 2 ( sin( θ))

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