By Barry Mazur
The e-book indicates how the paintings of mathematical imagining isn't as mysterious because it turns out. Drawing on quite a few creative assets the writer finds how a person can start to visualize the enigmatic 'imaginary numbers' that first baffled mathematicians within the sixteenth century.
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Extra info for Imagining Numbers: (Particularly the Square Root of Minus Fifteen)
O . . o o .. .. : .. .. 0 ... o . . 1 ... 0 0 .. . .. .. 0 ... 1 ... o . . 0 0 .. : ... ... 0 ... 0 . . 0 . . 1 0 0 ... O . . O . . O l jth (r - 1)th 1 0 0 .. 0 0 .. 0 0 column column It is easy to see that S - l = S and hence S-I A,-, S = SA,-, S will have the effect of interchanging the jth and (r - 1)th rows of A,-, S. This, of course, means that given A,-, we can write down S-lA,-, S without actually having to perform the matrix multiplications. ,nth rows of A,-, as we naturally require.
1 which had latent roots X, = 1, A When X = 1 we take U as Hence When A = 3 we get Note that A has only two linearly independent latent vectors. 3) we must, of course, take this into account in the above process. 4 is not quite so simple. If the latent wokors are required, having reached the stage of The Nethod of Danilevshy 49 in8i~itctof ju& r~dueiugDlto Frobenius form, it ia perhaps worth extending this reduction over the whole of A,-, so that we finish up with a matrix of the form from which it is fairly easy to determine the latent vectors of B.
0 . . 1 0 0 ... O . . O . . O l jth (r - 1)th 1 0 0 .. 0 0 .. 0 0 column column It is easy to see that S - l = S and hence S-I A,-, S = SA,-, S will have the effect of interchanging the jth and (r - 1)th rows of A,-, S. This, of course, means that given A,-, we can write down S-lA,-, S without actually having to perform the matrix multiplications. ,nth rows of A,-, as we naturally require. For this reason we cannot choose j > r - 1. 2 We cannot directly form C, because a,, interchange these two elements.