Inseparable Gershgorin discs and the existence of conjugate complex eigenvalues of real matrices

We investigate the converse of the known fact that if the Gershgorin discs of a real n-by-n matrix may be separated by positive diagonal similarity, then the eigenvalues are real. In the 2-by-2 case, with appropriate signs for the off-diagonal entries, we find that the converse is correct, which raises several questions. First, in the 3-by-3 case, the converse is not generally correct, but, empirically, it is frequently true. Then, in the n-by-n case, $ n\ge 3 $ n≥3, we find that if all the 2-by-2 principal submatrices have inseparable discs (‘strongly inseparable discs’), the full matrix must have a nontrivial pair of conjugate complex eigenvalues (i.e. cannot have all real eigenvalues). This hypothesis cannot generally be weakened.


Introduction
We assume standard Gershgorin disc theory as an inclusion region for the eigenvalues of an n-by-n matrix A = [a ij ], as reported in [1,Chaper 6].We also consider only real A, for reasons that will become clear.
Briefly, we review what Gershgorin's theorem says.Let ρ i ≡ n j=1,j =i |a ij |, 1 ≤ i ≤ n, denote the deleted absolute row sums of A. Then, every eigenvalue of A lies in at least one of the disks {z ∈ C : |z − a ii | ≤ ρ i }.Furthermore, a union of k of these n disks having no point in common with the remaining n−k disks contains precisely k eigenvalues of A. Notice that by applying the theorem to A T , a similar result is obtained using column sums of A, rather than row sums.Throughout this paper we will consider only the row version of the theorem.
It is known that if the Gershgorin discs for A can be separated by (positive) diagonal similarity, then the eigenvalues of A are all real.It is natural to ask if there is some sort of converse to this statement.Interestingly, there is.Observe that the similarity does not need to be diagonal or positive (any similarity can be applied), but we will specifically consider only positive diagonal similarities.
We will consider sign skew-symmetric (SSS) matrices.These are the real matrices A = [a ij ] such that a ij a ji < 0 for j = i.No assumption is made about the diagonal entries.Recall that, in the 2-by-2 case, which we first study, if the off-diagonal entries of the matrix are (weakly) of the same sign, the eigenvalues must be real and separability of the discs is not an issue.So, we only consider SSS matrices.
We first show that a 2-by-2 SSS matrix has a conjugate pair of complex eigenvalues if and only if its Gershgorin discs are (stricly) inseparable.If they are weakly inseparable (i.e. at best the two discs osculate), then there is a multiplicity 2 real eigenvalue (algebraic multiplicity 2, but geometric multiplicity 1).
This raises the question of what happens in the 3-by-3 case.Things are not as clean, but there are still interesting phenomena.Now, we may have two inseparable discs, with a third separated, or three inseparable discs (which may occur in two ways).In no case do inseparable discs guarantee a conjugate pair, but empirically counter examples are rare, especially with 3 overlapping discs in the strongest way.We give some simulation results to indicate this, as well as some indicative examples.
Another view of the 3-by-3 case is to not use the entire discs, but to consider the disc behaviour of the three 2-by-2 principal submatrices.Here, something very interesting happens.It can be that up to two principal submatrices can have conjugate pairs, without the entire 3-by-3 matrix having a conjugate pair, though this is also uncommon.But if all three have inseparable discs, then the full matrix must have a conjugate pair!Illuminating examples and implications are given.The 3-by-3 case may also be extended to the n-by-n case.

The 2-by-2 case
For simplicity, the expression 'has a nontrivial conjugate pair of complex (nonreal) eigenvalues' will often be replaced by 'has a conjugate pair'.Proof: (i) ⇐⇒ (ii).The discriminant of the characteristic polynomial of A is since bc < 0. Thus, A has a conjugate pair if and only if (a − d) 2 < 4|bc|.
(ii) ⇐= (iii).Without loss of generality, we may assume that D = 1 0 0 x , with x positive.Then, D −1 AD = a bx These discs properly overlap if and only if, for all x > 0, These two conditions can be expressed as follows: Complete the square to obtain Since we assume p(x) > 0, for all x > 0, and p(0) = |c| |b| > 0 by assumption, in particular, when x = |d−a| 2|b| ≥ 0, in particular for all x > 0, and thus the two Gershgorin discs of D −1 AD properly overlap.
In case (iii), we say that A has 'inseparable discs' and use the same terminology for larger n.

The 3-by-3 case
For a 3-by-3 real matrix A, the 3 Gershgorin discs may be ordered according to the numerical order of their centres.This order is not changed by diagonal similarity.The 'most separated' the Gershgorin picture might be (by positive diagonal similarity) can take a few forms (typically and up to reversal).
Of course, in the first case, all eigenvalues are real.In the other three cases, a conjugate pair may occur.
To seek sufficient conditions for the existence of a conjugate pair in a square SSS matrix A, here we consider the 2-by-2 principal submatrices with inseparable discs.We already know that if A is 2-by-2, one principal submatrix suffices.This raises the question of whether 1, 2 or 3 suffice in the 3-by-3 case (if any number do).Next, we present two insightful examples related to this question.
In connection to these examples, a useful observation is that via shifting, scaling and diagonal similarity, we can always normalize a real 3-by-3 SSS matrix A = [a ij ] with a nonzero diagonal (or its transpose or a permutation-similar matrix) into the form with a,b,c and d positive.In fact, assuming, for simplicity, that a 22 = a 11 , the shifting A ≡ A − a 11 I 3 creates a zero in the (1, 1) entry, the scaling A ≡ In each case, we also intended to record the ratio of matrices with a conjugate pair (i.e.not three real eigenvalues) using Matlab's built-in function isreal.
So, this experiment led us to admit that, in the 3-by-3 case, 3 2-by-2 principal submatrices with inseparable discs might be sufficient for the matrix to have a conjugate pair.
The following example is equally illuminating and reinforced the hypotheses posed by the previous example.

Example 3.2: Consider the 1-parameter 3-by-3 SSS family
Observe that −A(r) T = A(−r) and thus we may restrict our study to r ≥ 0. A calculation using the discriminant of the characteristic polynomial of A shows that if r ∈ [0, 2 √ 3), A(r) has a conjugate pair and for r not in this interval, all eigenvalues of A(r) are real.
Follows from Theorem 2.1 that for r < 4, the principal submatrices   has inseparable discs.So for r ∈ [2, 2 √ 3), A(r) has a conjugate pair and 2 principal submatrices with inseparable discs; for r ∈ [2 √ 3, 4), A(r) has real eigenvalues and also 2 principal submatrices with inseparable discs.See Figures 1 and 2.
Thus, these two examples show that 2 submatrices (or 1 submatrix) with inseparable discs are not sufficient for a conjugate pair.Three are.What happens in general?
Another question that we posed was: are the Gershgorin discs always separable by positive diagonal similarity when the eigenvalues are all real?The following analysis answers this question.( The three Gershgorin discs of Given that, for a > 0, the two inequalities (3) are always false for any x,y > 0. So, when r = 4, the matrix A has three distinct real eigenvalues but its Gershgorin discs are inseparable by positive diagonal similarity.
In general, given r ≥ 2 √ 3, for the discs of DAD −1 to be separable, we must require A symbolic calculation shows that if 2 √ 3 ≤ r ≤ 2 + 4 √ 2, the discs are inseparable by positive diagonal similarity; if r > 2 + 4 √ 2 the discs are separable.
So, in contrast to the 2-by-2 case, real eigenvalues is not a sufficient condition for the Gershgorin discs to be separable by positive diagonal similarity.

The n-by-n case using 2-by-2 principal submatrices Definition 4.1:
We say an n-by-n real matrix A is strongly inseparable if all its 2-by-2 principal submatrices have inseparable discs (a conjugate pair of complex eigenvalues).So, Theorem 2.1 says that strong inseparability suffices for a conjugate pair in the 2-by-2 case.
In the 3-by-3 case, 3 principal submatrices with inseparable discs suffice.To see this, we need some lemmas.Let p A denote the characteristic polynomial of a square matrix A and let A(i) denote the principal submatrix of A with row and column i deleted.Then (see [1, Chapter 1]), Lemma 4.2: If A is a square matrix over a field, then

Lemma 4.3: A monic quadratic polynomial has a conjugate pair if and only if it is unimodal with a positive minimum. In this event, its antiderivative (with zero constant term) also has a conjugate pair.
Proof: Let p(x) = x 2 + bx + c be a monic polynomial and = b 2 − 4c its discriminant.It is obvious that a real monic convex quadratic with a positive minimum must have < 0 as it cannot have real zeros, therefore it must have complex conjugate ones and therefore < 0. Since p is positive for all x, the cubic polynomial P(x) = 1 3 x 3 + b 2 x 2 + cx, antiderivative of p (with zero constant term), is monotone increasing and thus intersects the x-axis only once, i.e.P has one real zero.This zero must be simple given that a multiple zero (multiplicity 3) means that the derivative p has a double zero which contradicts the assumption.Thus P has one real zero and a conjugate pair of complex eigenvalues.Theorem 4.4: If a 3-by-3 SSS real matrix A is strongly inseparable, A has a conjugate pair of complex eigenvalues.
Proof: Since p A(i) are real monic convex quadratics, by Lemma 4.2, p A is obviously also a positive convex quadratic (non-monic) and therefore has no real zeros (but a conjugate pair).So, as in the proof of the second part of Lemma 4.3, the antiderivative of p A (with zero constant term) is monotone increasing and must have one real zero and a conjugate pair (otherwise p A could not have a conjugate pair).But p A (t) = thisantiderivative + det A must also have a conjugate pair since it is just a vertical translation of this antiderivative of p A .

Corollary 4.5: A 3-by-3 SSS matrix A with constant diagonal has a conjugate pair of complex eigenvalues.
Proof: Use Theorem 2.1 (ii) to conclude that a 3-by-3 SSS matrix A with constant diagonal is strongly inseparable.Remark 4.1: Notice that 'strongly inseparable' implies that every pair of discs overlaps for any diagonal similarity, but not conversely.Recall examples in Figures 1 and 2 in which every pair of discs overlap for any diagonal similarity but the matrix is not strongly inseparable (only two of the 2-by-2 principal submatrices have inseparable discs).
What about n-by-n real matrices, n > 3?
We emphasize that matrices with all the eigenvalues real and with 3 or more 2-by-2 principal submatrices with inseparable discs were rare or nonexistent.In absolute terms (on average), only 65 matrices (out of 16250 cases) with 3 2-by-2 principal submatrices with inseparable discs and no matrix (out of 6850 cases) with 4 2-by-2 principal submatrices with inseparable discs had all the eigenvalues real; and no cases of matrices with real eigenvalues and 5 or 6 2-by-2 principal submatrices with inseparable discs occured (out of 2950 and 1500 cases, respectively).See Figure 3.
So, this experiment predicts that in the 4-by-4 case, all the 6 2-by-2 principal submatrices with inseparable discs are almost certain to be sufficient for the matrix to have a conjugate pair.
Interestingly, Theorem 4.4 may be generalized, using the Gauss-Lucas theorem [2,3].Let Co(Rts(p)) denote the convex hull of the roots of the general complex polynomial p.Then, the roots of p ⊆ Co Rts(p) , or, equivalently,

Co Rts(p ⊆ Co Rts(p) . ( 6 )
A nice field of values proof of this classical fact was recently given in Ref. [4].Let A(i, j, . . ., k) denote the principal submatrix of A resulting from deletion of rows and columns i, j, . . ., k, and let A[i, j, . . ., k] denote the principal submatrix of A based on rows and columns i, j, . . ., k.We draw the reader's attention to the fact that we are using round and square brackets to denote different meanings.Let σ (A) denote the set of all the eigenvalues of A. Using the fact that p A = i p A(i) (see Lemma 4.2), we have the following Proof: Since each 2-by-2 has a conjugate pair of complex eigenvalues and its characteristic polynomial is positive and unimodal, so is the sum.Then, the sum has a conjugate pair, and, by Lemma 4.6, so does A. In fact, if all the eigenvalues of A were real, the inclusion (8) could not hold and thus A must have a conjugate pair.
Examples 3.1 and 3.2 show that the hypothesis cannot be weakened.
The following theorems state other sufficient conditions for the existence of a conjugate pair of eigenvalues for an n-by-n SSS matrix A can also be derived from Gauss-Lucas theorem.
Theorem 4.8: If A is an n-by-n SSS matrix, with n odd, in which each characteristic polynomial p A(i) takes on only positive values, then A has a conjugate pair of complex eigenvalues.

Theorem 2 . 1 :
Let A = a b c d , with bc < 0, i.e.A is SSS.Then, the following three statements are equivalent: (i) A has a conjugate pair; (ii) (a − d) 2 < 4|bc|; and (iii) for any positive diagonal matrix D, the two Gershgorin discs of D −1 AD properly overlap.

c x d
and the two Gershgorin discs of D −1 AD are z ∈ C : |z − a| ≤ |b|x and z ∈ C : |z − d| ≤ |c| x .

Example 3 . 3 :
Consider the matrix A(r) from the previous example and let r = 4 > 2 √3.The eigenvalues of A are the integers 1, 2 and 3. Define

Lemma 4 . 6 :
Gauss-Lucas to get each containment.We then have For any n-by-n real matrix A, Co σ (A) ⊇ Co This follows by running the above containments down to the 2-by-2 principal submatrices.generalization of Theorem 4.4 follows directly from Lemma 4.6.Theorem 4.7: If A is an n-by-n strongly inseparable SSS matrix (every 2-by-2 principal submatrix has a conjugate pair), then A has a conjugate pair.