Page 102 - Matrices theory and applications

5.4. Cyclic Matrices
are polynomial vector-valued functions, then the polynomial P(X):=
φ(V 1 (X),... ,V p (X)) has the derivative

P (X)= φ(V ,V 2 ,... ,V p )+ φ(V 1 ,V ,... ,V p )+ ··· + φ(V 1 ,... ,V p−1 ,V ).

2
1
One therefore has
1
2

P (X)= det(e ,a 2 ,... ,a n )+ det(a 1 , e ,... ,a n )+ ··· p 85
A
n
+ ··· +det(a 1 ,... ,a n−1 , e ),
1
n
where a j is the jth column of XI n − A and {e ,... , e } is the canonical
n
basis of IR . Developping the jth determinant with respect to the jth
column, one obtains
n

P (X)= P A j (X), (5.1)

A
j=1
where A j ∈ M n−1 (IR) is obtained from A by deleting the jth row and
the jth column. Let us now denote by B j ∈ M n (IR) the matrix obtained
from A by replacing the entries of the jth row and column by zeroes. This
matrix is block-diagonal, the two diagonal blocks being A j ∈ M n−1(IR)and
0 ∈ M 1 (IR). Hence, the eigenvalues of B j are those of A j , together with
zero, and therefore ρ(B j )= ρ(A j ). Furthermore, |B j |≤ A,but |B j | = A
because A is irreducible and B j is block-diagonal, hence reducible. It follows
(ρ(A)) is nonzero, with the
(Lemma 5.3.3) that ρ(B j ) <ρ(A). Hence P A j
in a neighborhood of +∞, which is positive. Finally,
same sign as P A j
P (ρ(A)) is positive and ρ(A)is a simple root.

A
This completes the proof of Theorem 5.3.1. A different proof of the sim-
plicity and another proof of the Perron–Frobenius theorem are given in
Exercises 2 and 4.
5.4 Cyclic Matrices
The following statement completes Theorem 5.3.1.
Theorem 5.4.1 Under the assumptions of Theorem 5.3.1, the set R(A) of
eigenvalues of A of maximal modulus ρ(A) is of the form R(A)= ρ(A)U p ,
where U p is the group of pth roots of unity, where p is the cardinality of
R(A). Every such eigenvalue is simple. The spectrum of A is invariant un-
der multiplication by U p .Finally, A is similar, by means of a permutation
n
of coordinates in IR , to the following cyclic form. In this cyclic matrix each
element is a block, and the diagonal blocks (which all vanish) are square