Patterns with Several Multiple Eigenvalues

Spec. Matrices 2014; 2:200–205

Research Article Open Access

J. Dorsey, C.R. Johnson, and Z. Wei* Patterns with several multiple eigenvalues

Abstract: Identified are certain special periodic diagonal matrices that have a predictable number ofpaired eigenvalues. Since certain symmetric Toeplitz matrices are special cases, those that have several multiple eigenvalues are also investigated further. This work generalizes earlier work on response matrices from circularly symmetric models.

Keywords: multiple eigenvalues, patterned matrix, special periodic diagonal matrices (PDM), Symmetric Toeplitz matrices

MSC: 15A18, 15B05, 15B57, 15B99

DOI 10.2478/spma-2014-0020 Received July 28, 2014; accepted December 2, 2014

1 Introduction

We consider n-by-n symmetric matrices with some simple equality relations among entries beyond those required by symmetry. This results in a coordinate subspace of n-by-n matrices that we view as a pattern. We are interested in patterns that guarantee several sets of multiple eigenvalues. Example. Let

⎡ 1 2 3 4 3 2 ⎤ ⎢ 2 7 2 6 4 6 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 3 2 1 2 3 4 ⎥ A = ⎢ ⎥ ⎢ 4 6 2 7 2 6 ⎥ ⎢ ⎥ ⎣ 3 4 3 2 1 2 ⎦ 2 6 4 6 2 7

Then, A has eigenvalues −3 (twice), 2 (twice), 3 and 23. In fact any matrix of the pattern

⎡ 1 1 1 ⎤ r0 r1 r2 r3 r2 r1 ⎢ r r2 r r2 r r2 ⎥ ⎢ 1 0 1 2 3 2 ⎥ ⎢ r1 r r1 r r1 r ⎥ R = ⎢ 2 1 0 1 2 3 ⎥ ⎢ 2 2 2 ⎥ ⎢ r3 r2 r1 r0 r1 r2 ⎥ ⎢ 1 1 1 ⎥ ⎣ r2 r3 r2 r1 r0 r1 ⎦ 2 2 2 r1 r2 r3 r2 r1 r0

1 2 1 2 (6 independent entries r0, r0, r1, r2, r2, r3) has two paired eigenvalues, as shall see. The matrix R may be written as a free linear combination of six 0, 1 matrices as follows:

*Corresponding Author: Z. Wei: Department of Mathematics, Imperial College London, UK, E-mail: [email protected] J. Dorsey: Department of Mathematics, Case Western Reserve University, Ohio, USA, E-mail: [email protected] C.R. Johnson: Department of Mathematics, College of William and Mary, Virginia, USA, E-mail: [email protected]

⎡ 1 0 0 0 0 0 ⎤ ⎡ 0 0 0 0 0 0 ⎤ ⎡ 0 1 0 0 0 1 ⎤ ⎢ 0 0 0 0 0 0 ⎥ ⎢ 0 1 0 0 0 0 ⎥ ⎢ 1 0 1 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 ⎢ 0 0 1 0 0 0 ⎥ 2 ⎢ 0 0 0 0 0 0 ⎥ ⎢ 0 1 0 1 0 0 ⎥ R = r0 ⎢ ⎥ + r0 ⎢ ⎥ + r1 ⎢ ⎥ ⎢ 0 0 0 0 0 0 ⎥ ⎢ 0 0 0 1 0 0 ⎥ ⎢ 0 0 1 0 1 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 0 0 0 0 1 0 ⎦ ⎣ 0 0 0 0 0 0 ⎦ ⎣ 0 0 0 1 0 1 ⎦ 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0

⎡ 0 0 1 0 1 0 ⎤ ⎡ 0 0 0 0 0 0 ⎤ ⎡ 0 0 0 1 0 0 ⎤ ⎢ 0 0 0 0 0 0 ⎥ ⎢ 0 0 0 1 0 1 ⎥ ⎢ 0 0 1 0 1 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1 ⎢ 1 0 0 0 1 0 ⎥ 2 ⎢ 0 0 0 0 0 0 ⎥ ⎢ 0 0 0 0 0 1 ⎥ +r2 ⎢ ⎥ + r2 ⎢ ⎥ + r3 ⎢ ⎥ ⎢ 0 0 0 0 0 0 ⎥ ⎢ 0 1 0 0 0 1 ⎥ ⎢ 1 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 1 0 1 0 0 0 ⎦ ⎣ 0 0 0 0 0 0 ⎦ ⎣ 0 1 0 0 0 0 ⎦ 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0

1 2 1 2 1 3 3 2 3 3 In order, we call these matrices R0, R0, R1, R2, R2 and R3. Their spectra are R0(0 , 1 ); R0(0 , 1 ); 2 2 1 2 3 2 2 3 3 3 R1(−2, −1 , 1 , 2); R2(−1 , 0 , 2) ; R2(−1 , 0 , 2); and R3(−1 , 1 ), with superscripts indicating multiplici- 1 2 1 2 ties. Interestingly, {R0, R0, R1, R2, R2, R3} does not form a commuting family, though most of the pairs do commute, and in the numerical example the eigenvalues (of the matrices with coefficients) do not add in any order. There is work on matrices with multiple eigenvalues, including much work on the possible multiplicity for symmetric matrices with a given graph e.g.[2], [3] , [5] , [6] and [4], as well as [8], in which some Toeplitz matrices with few distinct eigenvalues were given. We have been motivated, in part, by the observation [1] that the response matrices for certain circularly symmetric models have serveral paired eigenvalues. In fact, we will see here that the pairing has nothing to do with the fact that the underlying model has few parameters, but only to do with the pattern that results. This pattern has many more distinct entries, which may be taken to be independent. There is also some classical work on subspaces of matrices with multiple eigenvalues that seems to be largely unrelated [7]. In the next section, we introduce the notion of a special periodic diagonal matrix and then notice that for certain composite dimensions, special periodic diagonal matrices result from requiring that a symmetric matrix commute with a certain two special matrices. The example we mentioned is a special case. Then we show that these special PDM’s must have at least a certain number of paired eigenvalues (multiplicity 2 eigenvalues, including the possibility that some pairs may coincide). Since these special PDM’s include many symmetric Toeplitz matrices with palindromic first row off the diagonal, we then study Toeplitz matrices with multiple eigenvalues in further detail in the next section. These may occur for any size at least 3.

2 Periodic Diagonal Matrices with Paired Eigenvalues

In an m-by-m matrix C = (cij), for k = 0, 1,..., m − 1, the k-th diagonal is the set of entries {cij : j − i = k}. Except for diagonal 0 (the main diagonal), each diagonal is the union of two bands, one above the main diagonal and one below; we refer to these as bands k and −k. We are interested in matrices for which the entries on a diagonal form certain periodic sequences. The 6-by-6 pattern in the introduction is an example.

Let Qn,k be the n-by-n block circulant with k-by-k blocks ⎡ ⎤ 0 Ik 0 ... 0 ⎢ . . ⎥ ⎢0 0 I . . . ⎥ ⎢ k ⎥ ⎢ . . . . ⎥ ⎢ ...... ⎥ ⎢ . . 0⎥ ⎢ ⎥ ⎣0 0 ... 0 Ik⎦ Ik 0 0 ... 0 202 Ë J. Dorsey, C.R. Johnson, and Z. Wei

This is the nk-by-nk matrix that is the k-th power of the nk-by-nk basic circulant. If K is the “backward" identity matrix ⎡ ⎤ 0 ... 0 1 ⎢ . . ⎥ ⎢ . . . ⎥ ⎢ . 1 0⎥ K = ⎢ . ⎥ ⎢ . . . . . ⎥ ⎣0 . . . ⎦ 1 0 ... 0 [︃ ]︃ 1 0 let also Pm be defined as Pm = 0 K in which K is (m − 1)-by-(m − 1). We call an nk-by-nk matrix that is 1) symmetric;

2) commutes with Qn,k; and 3) commutes with Pnk an n, k special periodic matrix (PDM) matrix. For example,

⎡ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⎤ r0 r1 r2 r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 r2 r1 ⎢ r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 ⎥ ⎢ 1 0 1 2 3 4 5 6 7 7 6 5 4 3 2 ⎥ ⎢ 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 ⎥ ⎢ r2 r1 r0 r1 r2 r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 ⎥ ⎢ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⎥ ⎢ r r r r r r r r r r r r r r r ⎥ ⎢ 3 2 1 0 1 2 3 4 5 6 7 7 6 5 4 ⎥ ⎢ r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 ⎥ ⎢ 4 3 2 1 0 1 2 3 4 5 6 7 7 6 5 ⎥ ⎢ r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 ⎥ ⎢ 5 4 3 2 1 0 1 2 3 4 5 6 7 7 6 ⎥ ⎢ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⎥ ⎢ r r5 r4 r3 r2 r1 r0 r1 r2 r3 r4 r5 r r7 r7 ⎥ ⎢ 6 6 ⎥ ⎢ r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 ⎥ ⎢ 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 ⎥ ⎢ r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 ⎥ ⎢ 7 7 6 5 4 3 2 1 0 1 2 3 4 5 6 ⎥ ⎢ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⎥ ⎢ r6 r7 r7 r6 r5 r4 r3 r2 r1 r0 r1 r2 r3 r4 r5 ⎥ ⎢ ⎥ ⎢ r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 ⎥ ⎢ 5 6 7 7 6 5 4 3 2 1 0 1 2 3 4 ⎥ ⎢ r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 ⎥ ⎢ 4 5 6 7 7 6 5 4 3 2 1 0 1 2 3 ⎥ ⎢ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⎥ ⎢ r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 r2 r1 r0 r1 r2 ⎥ ⎢ 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 ⎥ ⎣ r2 r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 r2 r1 r0 r1 ⎦ 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 r1 r2 r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 r2 r1 r0 is a 5, 3 special PDM, while

⎡ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ⎤ r0 r1 r2 r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 r2 r1 ⎢ r1 r2 r2 r2 r2 r1 r2 r2 r2 r3 r1 r2 r2 r3 r3 ⎥ ⎢ 1 0 1 2 3 4 5 6 7 7 6 5 4 3 2 ⎥ ⎢ 1 2 3 3 2 1 2 3 3 2 1 2 3 3 3 ⎥ ⎢ r2 r1 r0 r1 r2 r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 ⎥ ⎢ 1 2 3 3 2 1 3 3 3 2 1 2 3 3 2 ⎥ ⎢ r r r r r r r r r r r r r r r ⎥ ⎢ 3 2 1 0 1 2 3 4 5 6 7 7 6 5 4 ⎥ ⎢ r1 r2 r2 r2 r2 r1 r3 r3 r2 r2 r1 r3 r2 r2 r2 ⎥ ⎢ 4 3 2 1 0 1 2 3 4 5 6 7 7 6 5 ⎥ ⎢ r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 ⎥ ⎢ 5 4 3 2 1 0 1 2 3 4 5 6 7 7 6 ⎥ ⎢ 1 2 2 3 3 1 2 2 2 2 1 2 2 2 3 ⎥ ⎢ r r5 r4 r3 r2 r1 r0 r1 r2 r3 r4 r5 r r7 r7 ⎥ ⎢ 6 6 ⎥ ⎢ r1 r2 r3 r3 r3 r1 r2 r3 r3 r2 r1 r2 r3 r3 r2 ⎥ ⎢ 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 ⎥ ⎢ r1 r2 r3 r3 r2 r1 r2 r3 r3 r2 r1 r3 r3 r3 r2 ⎥ ⎢ 7 7 6 5 4 3 2 1 0 1 2 3 4 5 6 ⎥ ⎢ 1 3 2 2 2 1 2 2 2 2 1 3 3 2 2 ⎥ ⎢ r6 r7 r7 r6 r5 r4 r3 r2 r1 r0 r1 r2 r3 r4 r5 ⎥ ⎢ ⎥ ⎢ r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 ⎥ ⎢ 5 6 7 7 6 5 4 3 2 1 0 1 2 3 4 ⎥ ⎢ r1 r2 r2 r2 r3 r1 r2 r2 r3 r3 r1 r2 r2 r2 r2 ⎥ ⎢ 4 5 6 7 7 6 5 4 3 2 1 0 1 2 3 ⎥ ⎢ 1 2 3 3 2 1 2 3 3 3 1 2 3 3 2 ⎥ ⎢ r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 r2 r1 r0 r1 r2 ⎥ ⎢ 1 3 3 3 2 1 2 3 3 2 1 2 3 3 2 ⎥ ⎣ r2 r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 r2 r1 r0 r1 ⎦ 1 3 3 2 2 1 3 2 2 2 1 2 2 2 2 r1 r2 r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 r2 r1 r0 is a 3, 5 special PDM. Both are 15-by-15. The former has 6 paired eigenvalues, while the latter has 5 paired eigenvalues. Notice that in each case, the entries on each diagonal form a period sequence, with period less than 15. As a result, there are many fewer potentially distinct entries than in a general symmetric matrix of the same Patterns with several multiple eigenvalues Ë 203 size. Also, the first row, off the diagonal is palindromic, and the structure of the resulting pattern subspace of Mn(R) could, as well, be described in terms of its bounds. An nk-by-nk symmetric matrix is an n,k special PDM if, for each 0 ≤ k ≤ m − 1 the i-th diagonal has a certain cadence that we describe below. Together with symmetry, and the fact that the first row is a palindrome off the diagonal, this allows one to construct the pattern foran n,k special PDM for arbitrary values of the parameters.

Claim 1. (Method for constructing a band of a special PDM) Let A be an nk-by-nk special PDM. Consider the i-th band of A. The band has period dividing k, so the first k j m entries determine the whole band. The band has length l = nk − i. These first k entries satisfy ri = ri where m ≡ l + 2 − jmodk. No other equalities necessarily hold.

′ Proof. An nk-by-nk matrix A = (aij) commutes with Qn,k if and only if aij = ai′ j′ whenever i ≡ i + kmodnk and j ≡ j′ + kmodnk. That is, the diagonals have period dividing k. This is clear due to the block circulant structure of Qn,k. Another straightforward verification shows that A commutes with Pnk if and only if the first row without its first entry is a palindrome, the first column without its first entry is a palindrome andthe (nk − 1) × (nk − 1) submatrix formed by deleting the first column and row of A has the property that its i-th row is its n − i-th row in reverse order.

That condition on the submatrix can be restated more concisely as aij = a(nk+2−i),(nk+2−j), whenever these indices are in the correct range. If we have an entry aj,j+i on the i-th band, then we can see aj,j+i = j a(nk+2−j),(nk+2−(j+i)) = a(nk+2−(j+i)),(nk+2−j) by symmetry. Noting that aj,j+i = ri and letting l = nk − i, we have j l+2−j ri = ri . Since the band has period dividing k, the j-th entry must equal the l + 2 − jmodk entry.

Example. Let A be the 15 × 15 periodic band matrix that commutes with Q3,5. We will construct the second 1 band. We have l = 15 − 2 = 13. The first entry we will call r2. For the second entry, 15 − 2 is not congruent 2 to 1 mod 5, so we call the second entry r2. For the third entry, 15 − 3 is congruent to 2 mod 5, so that entry 2 1 is also r2. For the fourth entry, 15 − 4 is congruent to 1 mod 5, so that entry is r2 again. Lastly, 15 − 5 is not 3 congruent to anything we already have, so that entry is called r2. Thus, the second band follows the pattern 1 2 2 1 3 r2, r2, r2, r2, r2, ie the entries in the 1,3; 2,4; 3,5; 4,6; and 5,7 positions.

n−1 Theorem 2. An n, k special PDM has ⌊ 2 ⌋k paired eigenvalues.

Proof. Follow the proof of Theorem 1 [1] (in slightly different notation) and notice that the origin of there- sponse matrix is not actually used, only its symmetry and commutativity properties.

The theorem appears to hold even if we relax the condition that the special PDM be a real matrix and instead allow for complex entries or even entries from finite fields. Note, however, that the proof of the theorem does not generalize in an obvious way, as it relied on the diagonalizability of real symmetric matrices, and for matrices over other fields, symmetry is not enough to guarantee diagonalizability.

3 Symmetric Toeplitz Matrices

Since Qn,k is the k-th power of the nk-by-nk basic circulant, an n, k special PDM with k = 1 is a circulant matrix, and all n,k special PDM’s are generalization of circulants. Every PDM with k = 1 is a symmetric 204 Ë J. Dorsey, C.R. Johnson, and Z. Wei toeplitz matrix. Let ⎡ ⎤ c0 c1 . . . cn−2 cn−1 ⎢c c c c ⎥ ⎢ n−1 0 1 n−2⎥ ⎢ . . . ⎥ ⎢ . . . . ⎥ C = ⎢ . cn−1 c0 . ⎥ ⎢ ⎥ ⎢ . . . . ⎥ ⎣ c2 . . c1 ⎦ c1 c2 . . . cn−1 c0 be an n-by-n circulant matrix. It is well-known that the eigenvalues of C are given by the following formula:

n−1 ∑︁ jk λj = ck ζ , j = 0, 1, ... , n − 1, k=0

2πi where ζ = exp( n ). Using this formula, it is easy to prove the theorem for this special case.

Claim 3. If cj = cn−j for j = 1, 2, ... , n − 1, then λj = λn−j for j = 1, 2, ... , n − 1.

Proof. We note two preliminary facts. First, if l = n − k, then k = n − l, and as k ranges from 1 to n − 1, l ranges from n − 1 to 1. Second, j(n − l) ≡ −jl ≡ (n − j)lmodn, and ζ n = 1, so ζ j(n−l) = ζ (n−j)l. Now we can easily see

n−1 n−1 n−1 n−1 ∑︁ jk ∑︁ jk ∑︁ j(n−l) ∑︁ (n−j)l λj = c0 + ck ζ = c0 + cn−k ζ = c0 + cl ζ = c0 + cl ζ = λn−j . k=1 k=1 l=1 l=1

Rather than generalizing to the special PDM case, the proof of this result suggests a different generalization to circulant matrices whose first row has a particular cadence related to the cycle shape of a permutation.

Claim 4. Let σ : Z/n → Z/n be given by g ↦→ mg, for m coprime to n. Then σ is a permutation on I = {0, 1, 2, ... , n −1}. Define an equivalence relation on I by j ∼ k if and only if j and k are in the same cycle when

σ is written as a product of disjoint cycles. If cj = ck whenever j ∼ k then λj = λk whenever j ∼ k.

s Proof. Let j ∼ k. Since j and k are in the same cycle of σ we have that there exists s ∈ Z/n such that j = m k. s s s lj σs (l)k s We therefore have lj = lm k for all l ∈ Z/n, so letting σ : l ↦→ m l, we have ζ = ζ . The map σ is an s s automorphism of Z/n, so summing over all σ (l) is the same as summing over all l. Since l and σ (l) are in the same cycle of σ we have cl = cσs (l). Now we can see that

∑︁ lj ∑︁ σs (l)k λj = cl ζ = cσs (l)ζ = λk . l∈I l∈I

The special case follows by taking m = n − 1 and noting that j(n − 1) ≡ n − jmodn, so that σ(j) = n − j.

Example. Take n = 9 and m = 2. Then σ written as a product of disjoint cycles is (0)(124875)(36).

Therefore, an n-by-n circulant matrix with c1 = c2 = c4 = c8 = c7 = c5 and c3 = c6 has an eigenvalue of multiplicity 1, an eigenvalue of multiplicity 6, and an eigenvalue of multiplicity 2.

Note that this does not generalize the work of Türen (see [8]), since those matrices have roughly half of their first row entries equal and not roughly half of their eigenvalues equal.

Acknowledgement: This work supported by NSF/DMS grant number DMS-0751964. Patterns with several multiple eigenvalues Ë 205

References

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