Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACES◮Find the transpose of the matrix( )3 1 2A =.0 4 1By interchanging the rows and columns of A we immediately obtain⎛A T = ⎝ 3 0⎞1 4 ⎠ . ◭2 1It is obvious that if A is an M × N matrix then its transpose A T is a N × Mmatrix. As mentioned in section 8.3, the transpose of a column matrix is arow matrix and vice versa. An important use of column and row matrices isin the representation of the inner product of two real vectors in terms of theircomponents in a given basis. This notion is discussed fully in the next section,where it is extended to complex vectors.The transpose of the product of two matrices, (AB) T , is given by the productof their transposes taken in the reverse order, i.e.This is proved as follows:(AB) T = B T A T . (8.39)(AB) T ij =(AB) ji = ∑ k= ∑ kA jk B ki(A T ) kj (B T ) ik = ∑ k(B T ) ik (A T ) kj =(B T A T ) ij ,and the proof can be extended to the product of several matrices to give(ABC ···G) T = G T ···C T B T A T .8.7 The complex and Hermitian conjugates of a matrixTwo further matrices that can be derived from a given general M × N matrixare the complex conjugate, denoted by A ∗ ,andtheHermitian conjugate, denotedby A † .The complex conjugate of a matrix A is the matrix obtained by taking thecomplex conjugate of each of the elements of A, i.e.(A ∗ ) ij =(A ij ) ∗ .Obviously if a matrix is real (i.e. it contains only real elements) then A ∗ = A.256