Page 97 - Textos de Matemática Vol. 39

SPECTRALLY ARBITRARY FACTORIZATION: THE NONDEROGATORY CASE 87
Lemma 1.7. Suppose that A ∈ M2(C) is a nonscalar matrix with eigenvalues λ1, λ2 and that α ∈ C is given. Then, there is a matrix similar to A with α in the 2, 2 position and nonzero entries in the 1, 2 and 2, 1 position if and only if α ̸= λ1,λ2.
Proof. The matrix A is similar to
λ1+λ2−α u vα
in which for any u ̸= 0,
Thescalarvisnonzero,unlessα=λ1 orα=λ2.
Lemma 1.8. Suppose that A ∈ Mn(C), n ≥ 3, is a nonscalar, nonsingular matrix and that α ∈ C is given. Then, there is a matrix similar to A with α in the n, n position and nonzero entries in the n − 1, n and n, n − 1 positions.
Proof. According to the prior lemma, we need only note that there is a sim- ilarity of A so that the 2-by-2 principal submatrix lying in the last two rows and columns does not have the eigenvalue α.
v= α(λ1 +λ2 −α)−λ1λ2. u
To complete the induction when the rank conditions are not strongly satisfied (rank(A − βiγiI) = 2 or 3 for some i) we consider the possible Jordan structures for A in this event. They are either
(1) nondiagonal: aI J4 (a), aI J3 (a) [b], aI J2 (a) J2 (b),
aI J2(a) [b] [c], aI J3(a), aI J2(a) [b], aI J2(b) or
(2) diagonal: aI [b] [c] [d], aI [b] [c].
Here, it is assumed a is among the list β1γ1, ..., βnγn and is different from b, c or d, but b, c, d or any subset may be equal. Of course, the block aI may be absent in dimension 4 in some cases. In each of the nondiagonal cases, a 3-by-3 principal submatrix may be found, to which Lemma 1.4 is applied. Via permutation similarity, this submatrix may be be placed in the lower right. Then an α may be chosen along with a similarity acting upon this lower right block, so that the induction hypothesis applies to the upper left (n−1)-by-(n−1) principal submatrix of the result. Because they held in the 3-by-3 submatrix, by identifying them as a direct summand, the nonvanishing almost principal minor conditions hold in the n-by-n matrix and they are undisturbed by the application of induction via a block similarity on the upper left (n−1)-by-(n−1) submatrix. Because of these, the partial special LU factorization delivered by induction extends to the entire matrix.
It remains to consider the diagonal cases (2). In this case it is more conve- nient to place a 3-by-3 matrix, that is not exceptional, in the upper left corner.