Remark 17

Suppose the conditions of No. 3 are satisfied; for every $\xi \in A$, denote by $|\xi|$ the element $\rho \geqslant 0$ of K such that $\rho^2 = \xi \overline{\xi}$ (absolute value of $\xi$). For every square matrix $M = (\alpha_{ij})$ over A, put $f(M) = \max_i |\alpha_{ii}|$, $g(M) = \max_{i,j} |\alpha_{ij}|$, and denote by $\varphi(M)$ the largest absolute value of the eigenvalues of $M$ (one will show that this definition has a meaning when A is the field of quaternions over K).

a) Let $A, B, D$ be three square matrices of order $n$ over A. Show that if $D$ is diagonal, one has

(use inequality (1) of No. 1). Deduce that (apply Prop. 5 to the hermitian matrix $BB^*$). Deduce that, for $m$ arbitrary square matrices $A_i$ ($1 \leq i \leq m$) over A, one has (For (3), reason by induction from (2). For (4), use Ex. 12; finally deduce (5) from (3).)

Show that inequality (3) no longer necessarily holds when $A_m^*A_m$ is replaced there by $A_mA_m^*$ (observe that one may have $f(A^*A) \neq f(AA^*)$), or when $\varphi(A_iA_i^*)$ is replaced by $f(A_iA_i^*)$ for $2 \leq i \leq m-1$ (take all the $A_i$ equal to the square matrix whose all elements are 1).

b) Let $u$ be a normal endomorphism of E; if $\lambda$ is an eigenvalue of $u$ (element of $K(i)$ when $A = K$), $\overline{\lambda}$ is an eigenvalue of $u^*u$. For every normal matrix $M$ (matrix of a normal endomorphism of E with respect to an orthonormal basis), one therefore has $\varphi^2(M) = \varphi(MM^*)$. Deduce that one also has $g(M) \leq \varphi(M)$, and more generally, if $M_1, \ldots, M_m$ are normal,

(use (4) and (5)).

c) Show that if $H$ is a positive hermitian matrix over A, one has $f(H) = g(H) \leq \varphi(H)$ (use (3) and b), by writing $H = AA^*$.