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By Rockswold G. K.

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Example text

We mention that the basic algebraic formulae involving complex numbers also hold for commuting matrices. A2n A2n 1 B C AB2n 1 C B2n /. 2 The binomial theorem for matrices. Let n 1 be an integer. C/ such that AB D BA, then ! A C B/n D Ak Bn k : k kD0 Proof The theorem can be proved by mathematical induction and by using the fact that Ak Bp D Bp Ak , for all k; p 2 N. 3 The set of complex numbers and matrices of order 2 In this section we establish an isomorphism between the field of complex numbers C and a special field of matrices of order 2.

R/ can be written as a linear combination of four orthogonal real matrices. 17 Complex orthogonal and skew orthogonal matrices. C/ such that AAT D I2 . C/ such that AAT D I2 . C/. 2A I 2 /2 D I 2 . 18 states that A is idempotent if and only if 2A involutory. C/ be nonzero idempotent matrices. 10). C/ be nonzero idempotent matrices. 11). C/ be such that A is idempotent and B is involutory. Prove that if A C B is involutory, then A D O2 . 22 A trace equality on special classes of matrices. C/ if and only if A D O2 .

41. A C B/T D AT C BT D A C B, which implies that A C B is a symmetric matrix. A C B/ and this implies that A C B is an antisymmetric matrix. R/. This implies that AT D A and AT D A. Thus 2A D O2 ) A D O2 . T T T (c) We have M D MCM C M 2M . R/. 42. A A/ is the imaginary part of A. A A / is the skew-Hermitian part of A. A/ are Hermitian matrices. 44. BC/A D A2 A D A3 . It follows that A3 D B3 D C3 . , 2 C C 1 D 0. Matrices A, A, and 2 A are distinct matrices which verify the conditions of the problem.

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A0-stable linear multistep formulas of the-type by Rockswold G. K.

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