By Paulus Gerdes
The aim of the e-book is to offer the reader a sense for the wonder and the surprises of mathematical study by means of increase step-by-step a idea of cycle matrices. The notions of matrix and cycle are offered for these readers who don't but comprehend the ideas. Then, beginning and experimenting with cycle matrices of low dimensions and fascinating the readers in a variety of actions, attractive effects are acquired with beautiful geometrical interpretation. progressively the consequences are prolonged to cycle matrices of upper dimensions after which generalised. A be aware is incorporated for these readers who want to use a working laptop or computer in exploring cycle matrices. appendices provide the prospect to additional use the pc through exploring inverses matrices and determinants.
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The aim of the e-book is to provide the reader a sense for the wonder and the surprises of mathematical examine by means of build up step-by-step a thought of cycle matrices. The notions of matrix and cycle are awarded for these readers who don't but comprehend the innovations. Then, beginning and experimenting with cycle matrices of low dimensions and interesting the readers in a number of actions, appealing effects are received with beautiful geometrical interpretation.
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8. 8 presents the addition of the matrices A and B that are both negative alternating cycle matrices of dimensions 6 by 6. In the resulting matrix A+B three cycles may be observed: in the first the numbers 3 and -1 alternate; in the second the numbers 1 and 8 and in the third the numbers -2 and 4. By consequence, the matrix A+B is a negative alternating cycle matrix of dimensions 6 by 6. The reader is invited to prove in general that the sum of any two negative alternating cycle matrices of dimensions 6 by 6 constitutes a negative alternating cycle matrix of dimensions 6 by 6.
2 3 4 1 3 1 -2 4 4 -2 1 3 1 4 3 -2 4 -3 2 -1 -3 -1 4 2 2 4 -1 -3 -1 2 -3 4 x x 4 -3 2 -1 -3 -1 4 2 2 4 -1 -3 -1 2 -3 4 -2 3 4 1 3 1 -2 4 4 -2 1 3 1 4 3 -2 = = -10 21 1 0 1 -10 0 21 21 0 -10 1 0 1 21 -10 -10 1 21 0 21 -10 0 1 1 0 -10 21 0 21 1 -10 20 Paulus Gerdes Once again, the diagonals of the matrices that result from the multiplication are equal and constant, whereas two alternating cycles appear, one in each matrix, which differ one unit in phase. 8 presents schematically the structure of the matrices resulting from the multiplication.
6). 6. ACTIVITIES • • How can one characterize the matrices resulting from the multiplication? Do they have a cyclic structure? Are the two matrices very different? 3, and multiply pairs of such matrices. What can be noted? 6, we conclude that both their principal and secondary diagonals are equal and composed of the repetition of the number 21 in the case of the principal diagonal and of the repetition of the number 4 in the case of the secondary diagonal. Moreover, the two matrices resulting from the multiplications have an alternating cycle composed of 3’s and 22’s.