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3 Berggren Basis and Completeness Relations
where I 2 is the identity matrix, and:
M
=
0 d
d c − b
Thus, the eigenvalues λ of M are equal to b + λ , with λ the eigenvalues of M .
If b = c, then M is proportional to the real symmetric matrix:
P =
0 1
1 0
and is thus trivially diagonalizable.
If b = c, then it is convenient to introduce the following matrix:
A =
0 a
a 2
.
Indeed
M = b I 2 + ((c − b)/2) A ,
with a = (2d)/(c − b), so that λ = b + ((c − b)/2) λ A , with λ A an eigenvalue of A.
The characteristic polynomial of matrix A is: P (X) = X 2 − 2X − a 2 , whose roots
are λ i = 1 − (−1) i √
1 + a 2 , i ∈ 1, 2.
The only 2 × 2 diagonalizable matrix with two identical eigenvalues is proportional to I 2 . Hence, one supposes that a = ±i. The normalized eigenvectors of A
thus read:
V 1 =
2(1 + a
2
+
1 + a 2 )
−1/2
1 +
√
1 + a 2
−a
and
V 2 =
2(1 + a
2
+
1 + a 2 )
−1/2
a
1 +
√
1 + a 2
,
V 1 and V 2 are always defined except when a = ±i.
Matrix A cannot be diagonalized only if a = ±i. In this latter case, |a| = 1, thus
of the same order of magnitude as the diagonal matrix element equal to 2 in A. In
this case (a = ±i), A has only one eigenvector which up to a constant factor equals:
V 0 =
±i
1
.
Its Berggren norm is equal to zero.