By Wilfred Kaplan

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**Additional resources for Advanced Calculus**

**Sample text**

Thus A is similar to the diagonal matrix B. 80). To handle the general case, it is best to extend the theory of matrices to the case of complex matrices, whose elements are complex numbers. All concepts generalize easily to this case, and, in particular, one can define complex eigenvalues and complex eigenvectors as above. For example, the matrix 37 Advanced Calculus. Fifth Edition + has the characteristic equation h2 1 = 0. T h e eigenvalues are 33,and associated eigenvectors are (2, 1 7 i). O n e proves, exactly as above, that when the matrix A has n distinct complex eigenvalues, A is similar 10 a diagonal matrix.

Then one defines the sum A B to be the m x n matrix C = ( c i j )such that cij = a,, b,, for all i and j ; that is, one adds two matrices by adding corresponding entries. For example, + + Let c be a number (scalar); let A = ( a i j )be an m x n matrix. Then one defines cA to be the m x n matrix B = ( b i j )such that bij = cai, for all i and j ; that is, cA 19 20 Advanced Calculus, Fifth Edition is obtained from A by multiplying each entry of A by c. For example, + We denote ( - l ) A by - A and B ( - A ) b y B - A.

7. Consider four equations in three unknowns: Assume that the pair of equations for i = 1 , 2 and the pair for i = 3 , 4 each represents a line in space as in Problem 6(a). Thus the solutions of the four equations represent the points common to two lines in space. Discuss the geometrical alternatives that can occur and relate them to the set of solutions of the given equations. 8. Let A be an m x n matrix. Let C = (cij) be the m x m matrix which differs from the identity only in that c h h = A # 0.

### Advanced Calculus by Wilfred Kaplan

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