Dr. Paulus Gerdes
Professor of Mathematics
Research Centre for Mathematics, Culture and Education
Postal address: C.P. 915, Maputo, Mozambique
E-mail: pgerdes@virconn.com
 
 

Introduction

The context in which cycle matrices were discovered and some of their basic symmetry properties are explained in Gerdes (2002a and 2005). The book Adventures in the World of Matrices (Gerdes 2004) presents an introduction to alternating cycle matrices and cycle matrices of any period p. The papers Gerdes (2002b and c) introduce helix and cylinder matrices and explore some of their relationships with cycle matrices.

In the following paper some further symmetry properties of alternating cycle matrices will be presented. The paper concludes with a geometrical interpretation of a class of permutation matrices that are simultaneously alternating cycle matrices. It is hoped that the article may underscore some of the beautiful symmetric properties of cycle matrices and may stimulate further exploration. 
 
 

Definition and generalities

Alternating cycle matrices constitute a particular class of square matrices. If m is an odd number (m = 2n+1), a matrix of dimensions m × m is called a positive alternating cycle matrix if all the elements on its principal diagonal are equal and if it is composed of n cycles of alternating numbers. Figure 1 illustrates the structure and general form of a positive alternating cycle matrix of dimensions 5×5.

If m is an odd number (m = 2n+1), a matrix of dimensions m × m is called a negative alternating cycle matrix if all the elements on its secondary diagonal are equal and if it is composed of n cycles of alternating numbers. Figure 2 illustrates the structure and general form of a negative alternating cycle matrix of dimensions 5×5.

Structure and general form of a negative alternating cycle matrix of dimensions 5×5

Figure 2

If m is an even number (m = 2n), a matrix of dimensions m × m is called a positive alternating cycle matrix if each of its diagonals is constant and if it is furthermore composed of n-1 cycles of alternating numbers. Figure 3 illustrates the structure and general form of a positive alternating cycle matrix of dimensions 6×6.

If m is an even number (m = 2n), a matrix of dimensions m × m is called a negative alternating cycle matrix if it is composed of n cycles of alternating numbers. Figure 4 illustrates the structure and general form of a negative alternating cycle matrix of dimensions 6×6.

The adjectives ‘positive’ and ‘negative’ are justified by the fact that the multiplication of positive and negative alternating cycles satisfies the same multiplication rules as positive and negative numbers: + × + = +, + × – = –, – × + = –, and – × – = +. Moreover, if A and B are two positive alternating cycle matrices of the same dimensions, then AB = BA. If C and D are two negative alternating cycle matrices of the same dimensions, then CD and DC are mutually symmetric: DC is the transposed matrix of CD. 

In the following some further symmetries of alternating cycle matrices will be presented.
 
 

Inverse matrices

Let A be a non-singular positive alternating cycle matrix of dimensions m × m. Let r = (b₁, …, b_m) be the first row of B = A^-1. We may construct the unique positive alternating cycle matrix C that has r as its first row. On its turn CA is a positive alternating cycle matrix, being its first row equal to the first row of BA. As B = A^-1, we have BA = A^-1A = I, where I represents the identity matrix of dimensions m ×m. The first row of BA is (1, 0, …, 0). By consequence, the first row of CA is (1, 0, …, 0). As CA is a positive alternating cycle matrix, its principal diagonal is composed of 1’s and its cycles composed of 0’s. It follows that CA = I, and thus C is the inverse matrix of A: C = B = A^-1. As C is by construction a positive alternating cycle matrix, we have proven that the inverse matrix of a non-singular positive alternating cycle matrix is a positive alternating cycle matrix itself.

The proof for the case that A is a non-singular negative alternating cycle matrix is analogous to the proof given. 
 
 

Determinants

By interchanging the 2^nd and the 3^rd row, the 4^th and the 5^th row, …, and the (2n)^th and the (2n+1)^th row, matrix P is transformed into matrix N. Interchanging two rows of a matrix implies changing the sign of the corresponding determinant. As we have a total of n interchanges, it follows that

Figure 5 illustrates the cases m=5 and m=6.

Two (positive or negative) alternating cycle matrices A and B may be called equivalent, if each row of A is a row of B, and vice versa. 

By interchanging several pairs of rows, matrix A can be transformed into any equivalent matrix B. By consequence, we have

Let A be a positive alternating cycle matrix of dimensions m × m. If we construct another positive alternating cycle matrix B that has the first row of A as one of its rows, what can be said about B?

Let us consider an example. F is a positive alternating cycle matrix of dimensions 5×5. We construct a positive alternating cycle matrix G such that the first row of F becomes the fourth row of G (see Figure 6).

Transformation of matrix F in matrix G

Figure 6

In this example we see that all rows of F appear as rows of G, and vice versa. The same holds for the columns of F and G. How can we explain this phenomenon? 

Let V₄ be the positive alternating cycle matrix of dimensions 5×5 that has 0’s on the principal diagonal, one cycle of 0’s and one alternating cycle of 0’s and 1’s such that the first element of its 4^th row is 1 (Figure 7). V₄ is a permutation matrix, i.e. a square matrix whose elements in any row (or any column) are all zero, except for one element equal to unity. 

Let us multiply the matrices V₄ and F. As the 4^th row of V₄ is (1, 0, 0, 0, 0), the 4^th row of V₄F is (a, b, c, d, e), that is the first row of F. As V₄ and F are positive alternating cycle matrices, the product is also a positive alternating cycle matrix. There exists only one positive alternating cycle matrix that has (a, b, c, d, e) as its 4^th row. It follows that V₄F = G. Similarly FV₄ = G. We may conclude that F and G are equivalent.

Based on this and similar experiences we may conjecture the following generalisation:

Let V_p be the positive alternating cycle matrix of dimensions m × m whose elements in the first column are all zero, except for the first element of the p^th row that is equal to unity. By consequence, V_p is a permutation matrix and its p^th row is (1, 0, …, 0). V_p A is a positive alternating cycle matrix and its p^th row is equal to the first row of matrix A. As there exists only one positive alternating cycle matrix that has the first row of A as its p^th row, it follows immediately that V_p A = B, and all rows of B are rows of A. We conclude that A and B are equivalent alternating cycle matrices.

Before considering in more detail the properties of the particular permutation matrices V_p, let us analyse the relationship between the inverse matrices of A and B in the case that A and B are non-singular.

We may conjecture

We have A V_p = V_p A = B, for a certain p = 2, …, m. The inverse of V_p is another matrix of the same type V_q (cf. Gerdes 2005). We have

and

Thus A^-1 V_q = V_q A^-1 = B^-1. In other words, the matrices A^-1 and B^-1 are equivalent positive alternating cycle matrices. 
 
 

Permutations

The matrices V_p constitute a particular class of permutation matrices: the permutation matrices that are simultaneously positive alternating cycle matrices. 

Let us observe the example V₄ when m = 5 (Figure 7). Multiplying a positive alternating cycle matrix of dimensions 5×5 by V₄, transforms the first row of A into the 4^th row of the new matrix B (= A V₄), the 4^th row of A becomes the 3^rd of B, the 3^rd row of A becomes the 2^nd of of B, the 2^nd row of A becomes the 5^th of B, and the 5^th row of A becomes the 1^st of B. In other words, the cycle notation for the permutation of the rows is (1 4 3 2 5). 

Similarly, the matrices V₂, V₃, and V₅ of dimensions 5×5, correspond to (1 2 4 5 3), (1 3 5 4 2), and (1 5 2 3 4).

How can we understand and interpret the particular sequence of the numbers 1 to 5 in the cycle notation of these permutations?

In general, any cycle of a cycle matrix passes exactly twice through each row. In the particular case of V₄, the non-zero cycle passes exactly twice to each of the five rows. Correspondingly, let us mark 10 dots at equal intervals around a circle and number these dots twice, on the left and on the right, from 1 to 5, as illustrated in Figure 8.

The permutation (1 4 3 2 5) can now be represented by a polygon: starting with the 1-dot on the left side, advancing clockwise four dots one arrives at the 4-dot on the right side (draw the corresponding segment 1-4); advancing another four dots one arrives at 3-dot on the left side (draw the corresponding segment 4-3); advancing once more four dots, one arrives at the 2^-dot on the right (segment 3-2), and then at the 5-dot on the left (drawing the segment 2-5) and finally by advancing four more dots one returns to the initial 1-dot on the left (Figure 9). The final representation of the permutation (1 4 3 2 5) is that of a regular pentagram.

The inverse permutation of (1 4 3 2 5) is (5 2 3 4 1) that may be written as (1 5 2 3 4), and its representation is the vertical mirror image of the first (Figure 10).

On the one hand, the mutually inverse permutations (1 4 3 2 5) and (1 5 2 3 4) correspond to the positive alternating cycle matrices V₄ and V₅ with V₄V₅ = V₅ V₄ = I. At the same time these matrices are mutually symmetrical: a reflection in the principal diagonal of V₄ transforms V₄ into V₅, and vice versa (Figure 11). In other words, V₅ is the transposed matrix of V₄: V₅ = (V₄)^T. On the other hand, the polygonal-circle representations of these matrices (Figures 8 and 10) are mutually symmetrical too.

The sum V₄ + V₅ is a positive cycle matrix of period 1 (all cycles are constant), with two axes of symmetry. 

Similarly, the mutual inverse permutations (1 2 4 5 3) and (1 3 5 4 2), corresponding to V₂ and V₃, can be represented by regular pentagons (Figure 12).

We are now in a position to extrapolate and to formulate a general conjecture. Its proof follows directly from the understanding of the reflection of the branches of any alternating cycle in the sides of the square circumscribed to the matrix. The theorem may be formulated as follows:

References
 

Gerdes, Paulus (2002a), From Liki-designs to cycle matrices: The discovery of attractive new symmetries, Visual Mathematics, Vol. 4, No. 1, (http://members.tripod.com/vismath7/gerd/)

Gerdes, Paulus (2002b), Helix matrices, Visual Mathematics, Vol. 4, No. 2, June 2002 (http://members.tripod.com/vismath8/gerdhel/hel.htm)

Gerdes, Paulus (2002c), Cylinder matrices, Visual Mathematics, Vol. 4, No. 2, June 2002 (http://members.tripod.com/vismath8/gerdcyl/cyl1.htm)

Gerdes, Paulus (2004), Aventuras no Mundo das Matrizes [Adventures in the World of Matrices], book manuscript

Gerdes, Paulus (2005), Mathematical research inspired by African cultural practices: The example of mirror curves, Lunda-designs and related concepts, in: Giandomenico Sica (Ed.), What Mathematics from Africa?, Polimetrica, Milano, 2005, 29-47.