Exercise 9, § 6

(a) Let $n$ be an integer and let M be a square matrix with coefficients in $\mathbf{N}$. Suppose that $\varrho (M^tM) =\varrho ($M$)^2$ and that $|\varrho ($M$)|<2$. There exists an integer $m\geqslant 2$ such that $\varrho ($M) = 2 cos($\pi /m$). (Use Kronecker's theorem, cf. A, V, p. 155, Exercise 17.)

(b) For every integer $m\geqslant 2$, there exists a square matrix M with coefficients in $\mathbf{N}$ such that $\varrho ($M) = 2 cos($\pi /m$).

(c) Let $(A,\mathscr{B})$ be a natural unital basal algebra of finite rank. Let $b\in \mathscr{B}$. If pf($b$)$<2$, then then there exists an integer $m\geqslant 3$ such that pf($b$) $= 2$ cos($\pi /m$).

Exercises 10 to 24 propose another proof of Corollary 5 of III, p. 100 as well as certain refinements. The following notions will be used.

Let I be a finite set and let $n$ be its cardinal.

– We denote by $1_I$ the real unit matrix of type $(I,I)$.

– A square matrix of type $(I,I)$ with real coefficients is said to be positive if all its coefficients are $\geqslant 0$ and strictly positive if all its coefficients are $>0$.

– A square matrix $(a_{ij})_{i,j\in I}$ of type $(I,I)$ with real coefficients is said to be reducible if there exists a partition of I into two non-empty subsets $I'$ and $I''$ such that $a_{ij}= 0$ for all $(i, j)\in I'\times I''$; it is said to be irreducible if it is not reducible.

– One calls coordinate subspace of $\mathbf{R}^I$ any vector space subspace of $\mathbf{R}^I$ generated by a subset of vectors of the canonical basis. There are $2^n$ coordinate subspaces; for each integer $k\in \{0, . . . , n\}$, those among them which are of dimension $k$ are in number $(^n_k)$.

A real matrix of type $(I,I)$ is therefore irreducible if and only if the only coordinate subspaces which it stabilizes are $\{0\}$ and $\mathbf{R}^I$.

For all $x$ and $y$ in $\mathbf{R}^I$ we shall write $x\preccurlyeq y$ (and $y\succcurlyeq x$) if $y-x$ is in the cone of elements of $\mathbf{R}^I$ with positive coordinates, that is to say if $x_i\leqslant y_i$ for all $i\in I$. We shall write $x\prec y$ (and $y\succ x$) if all the coordinates of $y-x$ are strictly positive.