By Charles G. Cullen
Undergraduate-level creation to linear algebra and matrix conception bargains with matrices and linear platforms, vector areas, determinants, linear differences, similarity, polynomials, and polynomial matrices. additionally spectral decomposition, Jordan canonical shape, resolution of the matrix equation AX=XB, and over 375 difficulties, many with solutions. "Comprehensive." — digital Engineer's layout journal.
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Additional info for Matrices and Linear Transformations: Second Edition
8. 7. 9. For evaluate m(S). 10. For evaluate p(T). Verify that the diagonal entries of p(T) are p(1), p(– 1), and p(3). 5 TRANSPOSITION If A is an n × m matrix, then the m × n matrix obtained from A by interchanging its rows and columns is called the transpose of A and is denoted by AT. 10 If A is n × m over , then AT is the m × n matrix defined by entij(AT) = entji(A). For example, if then We will find it convenient and space-saving to write column matrices as transposed row matrices. You will see later that transposition is a useful and important operation on matrices which will allow us to deduce column analogs of results about the rows of a matrix and vice versa.
The following theorem lists these properties formally. It is assumed that all matrices written are conformable for addition. 5 Matrix addition has the following properties: I. Commutative: that is, A + B = B + A. II. Associative: that is, A + (B + C) = (A + B) + C. III. There is an identity element for matrix addition. ) IV. There is an additive inverse (negative): that is, for each matrix A, there is a unique matrix – A satisfying A + (– A) = Z. V. Cancellation Law: that is, if A + B = A + C, then B = C.
Now, by the definition of the transpose and the definition of matrix multiplication Thus (AB)T = BTAT as claimed. IV. See Exercise 3. V. AT(A– 1A)T = (A– 1 A)T = IT = I by III and, similarly, (A– 1)TAT = (AA– 1)T = I. Thus (AT)– 1 = (A– 1)T as claimed. The definition of the transpose of a matrix makes perfectly good sense independent of the nature of the elements of the matrix. For matrices whose elements are complex numbers, there is a related idea which turns out to be more useful in practice.