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 cir- cularly 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 2 1 2 3 4 3 2 3 6 2 7 2 6 4 6 7 6 7 6 7 6 3 2 1 2 3 4 7 A = 6 7 6 4 6 2 7 2 6 7 6 7 4 3 4 3 2 1 2 5 2 6 4 6 2 7 Then, A has eigenvalues −3 (twice), 2 (twice), 3 and 23. In fact any matrix of the pattern 2 1 1 1 3 r0 r1 r2 r3 r2 r1 6 r r2 r r2 r r2 7 6 1 0 1 2 3 2 7 6 r1 r r1 r r1 r 7 R = 6 2 1 0 1 2 3 7 6 2 2 2 7 6 r3 r2 r1 r0 r1 r2 7 6 1 1 1 7 4 r2 r3 r2 r1 r0 r1 5 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] © 2014 J.Dorsey et al., licensee De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. Patterns with several multiple eigenvalues Ë 201 2 1 0 0 0 0 0 3 2 0 0 0 0 0 0 3 2 0 1 0 0 0 1 3 6 0 0 0 0 0 0 7 6 0 1 0 0 0 0 7 6 1 0 1 0 0 0 7 6 7 6 7 6 7 6 7 6 7 6 7 1 6 0 0 1 0 0 0 7 2 6 0 0 0 0 0 0 7 6 0 1 0 1 0 0 7 R = r0 6 7 + r0 6 7 + r1 6 7 6 0 0 0 0 0 0 7 6 0 0 0 1 0 0 7 6 0 0 1 0 1 0 7 6 7 6 7 6 7 4 0 0 0 0 1 0 5 4 0 0 0 0 0 0 5 4 0 0 0 1 0 1 5 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 2 0 0 1 0 1 0 3 2 0 0 0 0 0 0 3 2 0 0 0 1 0 0 3 6 0 0 0 0 0 0 7 6 0 0 0 1 0 1 7 6 0 0 1 0 1 0 7 6 7 6 7 6 7 6 7 6 7 6 7 1 6 1 0 0 0 1 0 7 2 6 0 0 0 0 0 0 7 6 0 0 0 0 0 1 7 +r2 6 7 + r2 6 7 + r3 6 7 6 0 0 0 0 0 0 7 6 0 1 0 0 0 1 7 6 1 0 0 0 0 0 7 6 7 6 7 6 7 4 1 0 1 0 0 0 5 4 0 0 0 0 0 0 5 4 0 1 0 0 0 0 5 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, fR0, R0, R1, R2, R2, R3g 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 symmet- ric 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 fcij : j − i = kg. 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 2 3 0 Ik 0 ... 0 6 . 7 60 0 I . 7 6 k 7 6 . 7 6 . 7 6 . 07 6 7 40 0 ... 0 Ik5 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 2 3 0 ... 0 1 6 . 7 6 . 7 6 . 1 07 K = 6 . 7 6 . 7 40 . 5 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, 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 r0 r1 r2 r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 r2 r1 6 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 7 6 1 0 1 2 3 4 5 6 7 7 6 5 4 3 2 7 6 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 7 6 r2 r1 r0 r1 r2 r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 7 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 6 r r r r r r r r r r r r r r r 7 6 3 2 1 0 1 2 3 4 5 6 7 7 6 5 4 7 6 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 7 6 4 3 2 1 0 1 2 3 4 5 6 7 7 6 5 7 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 7 6 7 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 6 r r5 r4 r3 r2 r1 r0 r1 r2 r3 r4 r5 r r7 r7 7 6 6 6 7 6 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 7 6 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 7 6 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 7 6 7 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 6 r6 r7 r7 r6 r5 r4 r3 r2 r1 r0 r1 r2 r3 r4 r5 7 6 7 6 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 7 6 5 6 7 7 6 5 4 3 2 1 0 1 2 3 4 7 6 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 r1 r2 r2 7 6 4 5 6 7 7 6 5 4 3 2 1 0 1 2 3 7 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 6 r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 r2 r1 r0 r1 r2 7 6 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 7 4 r2 r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 r2 r1 r0 r1 5 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 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 r0 r1 r2 r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 r2 r1 6 r1 r2 r2 r2 r2 r1 r2 r2 r2 r3 r1 r2 r2 r3 r3 7 6 1 0 1 2 3 4 5 6 7 7 6 5 4 3 2 7 6 1 2 3 3 2 1 2 3 3 2 1 2 3 3 3 7 6 r2 r1 r0 r1 r2 r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 7 6 1 2 3 3 2 1 3 3 3 2 1 2 3 3 2 7 6 r r r r r r r r r r r r r r r 7 6 3 2 1 0 1 2 3 4 5 6 7 7 6 5 4 7 6 r1 r2 r2 r2 r2 r1 r3 r3 r2 r2 r1 r3 r2 r2 r2 7 6 4 3 2 1 0 1 2 3 4 5 6 7 7 6 5 7 6 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 7 6 7 6 1 2 2 3 3 1 2 2 2 2 1 2 2 2 3 7 6 r r5 r4 r3 r2 r1 r0 r1 r2 r3 r4 r5 r r7 r7 7 6 6 6 7 6 r1 r2 r3 r3 r3 r1 r2 r3 r3 r2 r1 r2 r3 r3 r2 7 6 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 7 6 r1 r2 r3 r3 r2 r1 r2 r3 r3 r2 r1 r3 r3 r3 r2 7 6 7 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 6 1 3 2 2 2 1 2 2 2 2 1 3 3 2 2 7 6 r6 r7 r7 r6 r5 r4 r3 r2 r1 r0 r1 r2 r3 r4 r5 7 6 7 6 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 r1 7 6 5 6 7 7 6 5 4 3 2 1 0 1 2 3 4 7 6 r1 r2 r2 r2 r3 r1 r2 r2 r3 r3 r1 r2 r2 r2 r2 7 6 4 5 6 7 7 6 5 4 3 2 1 0 1 2 3 7 6 1 2 3 3 2 1 2 3 3 3 1 2 3 3 2 7 6 r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 r2 r1 r0 r1 r2 7 6 1 3 3 3 2 1 2 3 3 2 1 2 3 3 2 7 4 r2 r3 r4 r5 r6 r7 r7 r6 r5 r4 r3 r2 r1 r0 r1 5 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.