Page 96 - Textos de Matemática Vol. 39

86 CHARLES R. JOHNSON AND YULIN ZHANG
in case (1) and the same, with b = 1, in case (2). Of course, B[1, 2] necessarily has an eigenvalue 1, so that its other eigenvalue is α (accounting for one of the restrictions). These matrices B complete the proof.
For general n, we wish to show the following
Theorem 1.5. Let A ∈ Mn be nonsingular and nonscalar and suppose that
n
β1,...,βn,γ1,...γn ∈ C are given so that βiγi = detA. Then A is similar
i=1
to a matrix with special LU factorization for β1, ..., βn, γ1, ...γn if and only if
there is no i, 1≤i≤n, such that rank(A−βiγiI)=1.
Proof. Of course, the case n = 2 is in Lemma 1.2; the case n = 3 is an easy corollary to Lemma 1.4 by taking k = 3, α1 = β1γ1, α2 = β2γ2, α3 = β3γ3 and α = β1γ1β2γ2. Note that the restrictions in the lemma, in the not nonderogatory case are consistent with these requirements. Then, the 2-by-2 case may be applied to the upper left block, without disturbing the nonzero almost principal minor conditions that ensure that the LU factorization will be special. Using the determinant condition, the product of the 3, 3 entries of L and U will be β3γ3 and these entries can be chosen separately as β3 and γ3 without loss of the generalities. This gives
Corollary 1.6. Let A ∈ M3(C) be nonsingular and nonscalar and suppose that 3
We now turn to a proof of Theorem 1.5. The overall strategy is an induc- tion on n. Lemma 1.1 and Corollary 1.6 supply the initial cases n = 2, 3. At the induction step, we need only show that the requirements on A have been transferred to the upper left (n − 1)-by-(n − 1) principal submatrix of some similarity of A, in which the last two almost principal minors are nonzero. It is easy to observe that when induction is applied by a block similarity acting upon the upper left submatrix, the nonzero, almost principal minor condition will not be altered. This ensures that the special LU factorization will extend.
In the event that the rank conditions, rank(A − βiγiI) > 1, are strongly satisfied: mini rank(A − βiγiI) > 3, they are obviously conveyed by standard rank inequalities to the upper left submatrix. Then, by observing the relation- ship between the minors of B and its inverse, the proof is easily completed in this case, using the following two lemmas.
β1, β2, β3, γ1, γ2, γ3 ∈ C are given so that βiγi = det A. Then, A is similar i=1
to a matrix with special LU factorization for β1, β2, β3, γ1, γ2, γ3 if and only if there is no i, 1 ≤ i ≤ n, such that rank(A − βiγiI) = 1.