Mathematical Methods for Physics and Engineering - Matematica.NET

8.5 FUNCTIONS OF MATRICESThe identity matrix I has the propertyAI = IA = A.It is clear that, in order for the above products to be defined, the identity matrixmust be square. The N × N identity matrix (often denoted by I N )hastheform⎛⎞1 0 ··· 0.I N =0 1 .⎜ .⎝. ⎟. .. 0 ⎠ .0 ··· 0 18.5 Functions of matricesIf a matrix A is square then, as mentioned above, one can define powers of A ina straightforward way. For example A 2 = AA, A 3 = AAA, or in the general caseA n = AA ···A(n times),where n is a positive integer. Having defined powers of a square matrix A, wemay construct functions of A of the formS = ∑ na n A n ,where the a k are simple scalars and the number of terms in the summation maybe finite or infinite. In the case where the sum has an infinite number of terms,the sum has meaning only if it converges. A common example of such a functionis the exponential of a matrix, which is defined byexp A =∞∑n=0A nn! . (8.38)This definition can, in turn, be used to define other functions such as sin A andcos A.8.6 The transpose of a matrixWe have seen that the components of a linear operator in a given coordinate systemcan be written in the form of a matrix A. We will also find it useful, however,to consider the different (but clearly related) matrix formed by interchanging therows and columns of A. The matrix is called the transpose of A and is denotedby A T .255