By Eckhard M.S. Hitzer, University of Fukui, Prof. Ryosuke Nagaoka (University of the Air, Japan)
Yes, please have a look at the pattern. So here on this pattern you see a polynomial f, which is x squared minus x plus one, times a polynomial g, which is x squared plus x plus one. And as you know from your highschool days this is x to the power of four plus x squared plus one. And now we can replace x in each occurance by a matrix A. So we have first matrix A squared, minus matrix A, plus the unit matrix E, times again A squared plus A plus E. And this equals also A to the power of four, plus A squared, plus the unit matrix E.
Now I show you another example. And here we will have two polynomials f and g of two transcentdental elements x and y. And the product of the two polynomials is another polynomial p again of x and y. Now we can replace x by A, a matrix and y by B, another matrix. And we can ask if the product of these two matrix polynomials also equals the product polynomial p of the matrices A and B, just replacing x and y also in the result.
Here you see a concrete example. f is x minus y, and g is x plus y. And as you know again, this is x squared minus y squared. So let us also replace here x by A and y by B. So f of A and B is A minus B times g (this is A plus B). And we can ask, if this is also A squared minus B squared. But now I will show you in a calculation, that this actually does not hold.
So here you see f (=) A plus B times g (=) A minus B, and this equals A times A is A squared, plus A times minus B is minus A B, plus B times A is B A, plus B times minus B is minus B squared. And here we have now the term minus A B plus B A, but because in general matrix multiplication does not commute, these two terms do not cancel. And therefore in this formula A minus B times A plus B is not A squared minus B squared, there are additional terms.
Soli Deo Gloria. Created by
Eckhard Hitzer (Fukui).
Last Modified 16 January 2004
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