Exercise 14, § 6

Let A be a real square matrix of type $(I,I)$, positive and irreducible. We take up again the notation of Exercise 12.

Let $(\lambda_h)_{h\in H}$ be the family of complex eigenvalues of A whose modulus is $r$, repeated with their multiplicities; set $v_h=\lambda_h/r$ for every $h\in H$.

Let $h\in H$; by Exercise 13 applied to C = A and $\gamma =\lambda_h$, there exists a matrix $D_{(h)}$ such that A $=v_hD_{(h)}$AD$_{(h)}^{-1}$ and $D_{(h)}^+= 1_I$.

a) Let $z\succ 0$ be an extremal vector of A. For $h\in H$, the vector $y_{(h)}=D_{(h)}z$ is an eigenvector of A for the eigenvalue $\lambda_h$. b) Now prove that for $h\in H$, the eigenvalue $\lambda_h$ of A is simple. Deduce that the matrices $D_{(h)}$ and the vectors $y_{(h)}$ are uniquely determined up to a scalar factor.

c) Now prove that, for every $h$ and $k$ in $H^2$, one has A $=u_hu^{-1}_kD_{(h)}D_{(k)}^{-1}$AD$_{(k)}D_{(h)}^{-1}$ and therefore that the vector $D_{(h)}D_{(k)}^{-1}z$ is an eigenvector of A associated with the eigenvalue $u_hu^{-1}_kr$.

d) Deduce that $\{u_h\}_{h\in H}$ is a subgroup of $\mathbf{C}^*$ and that there exists a family $(\mu_h)_{h\in H}$ of nonzero complex numbers such that $\{\mu^{-1}_hD_{(h)}\}_{h\in H}$ is a subgroup of the group of invertible matrices of type $(I,I)$ with complex coefficients, and that these subgroups are cyclic of order Card(H). (Choose an index $i\in I$ and normalize $D_{(h)}$ so that its coefficient with row and column indices $i$ is equal to 1.)

In the remainder of the exercise, it is supposed that $H =\mathbf{Z}/d\mathbf{Z}$ and that there exists $\omega \in \mathbf{C}^*$ such that $\omega^d= 1$ and such that $u_h=\omega^h$ for $h\in H$. Setting D = $D_{(1)}$, one then has $D^d= 1_I$ and $D_{(h)}=D^h$ for every $h\in H$.

e) Deduce that A $=\omega$DAD$^{-1}$.

Consider the partition $(I_h)_{h\in H}$ of I, such that for $i\in I$, the coefficient with index $(i, i)$ of D is equal to $\omega^h$ if and only if $i\in I_h$. For every $h$, set $n_h=$ Card(I$_h$). The matrix D is written "in blocks"

D = . ..

and we consider the corresponding decomposition of A:

A = ... ... . .. ...,

so that for $h, k\in H,A_{h,k}$ is a matrix of type $(I_h,I_k)$.

f) For all $(h, k)\in H$, one has $\omega A_{h,k}=\omega^{k-h}A_{h,k}$. Deduce that $A_{h,k}= 0$ if $k-h\not = 1$.