Matrix Theory and Linear Algebra - download pdf or read online
By I. N. Herstein
This can be a attractive, no longer so recognized textual content of the bright writer I.N. Herstein.
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This necessitates the enlargement of the concept of real number to some wider arena: that of complex numbers. We do this in the next section. When this has been accomplished, we shall not have to worry about the presence or absence of roots for the polynomials that will arise. , a. ). · · · For the moment, let's act as if the polynomials we are dealing with had real roots-something that is often false. At any rate, let's see where this will lead us. Definition. A characteristic root of A is a number a such that PA(a) = 0.
Do not confuse f-1(s) with (f(s))-1 = : = � � s - s = � 5 for every s in IR. " In our example, f-1(s) f s) s = � 5, 32 The 2 Matrices 2 x (f (s)r1 whereas . mstance, f-1(1) = = [Ch. 1 1 f (s) = 1 -5 -6 = 1 and these are not equal as functions on S. For 6s + 5 -4 - while (f(l))-1 6 = - 1 6 l - + 5 = -. Let Sbe the set of all positive real numbers and suppose that S-+ Sis defined by f: f (s) l/s. = Clearly, f is a 1 mapping of Sonto itself, -1 hence 1-1 exists. What is f-1? We leave it to the reader to prove that 1 /s = s f( ) for all s, so that 1-1 = f-1(s) = f.
B) AB+ BA is symmetric. (c) (d) AB - BA is skew-symmetric. ABA is symmetric. If A is invertible, then A-1 is symmetric. (e) 12. If A is skew-symmetric, show that A" is skew-symmetric if n is a positive odd integer and A" is symmetric if n is a positive even integer. 13. If A and B are skew-symmetric, show that • (a) AB + BA is symmetric. (b) AB - BA is skew-symmetric. (c) A(AB+ BA) - (AB+ BA) A is symmetric. (d) ABA is skew-symmetric. (e) If A is invertible, then A-1 is skew-symmetric. 14. , AB= BA) show that (a) If A is symmetric and B skew-symmetric, then AB is skew-symmetric.
Matrix Theory and Linear Algebra by I. N. Herstein