Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.4 BASIC MATRIX ALGEBRA Now, sincexis arbitrary, we can immediately deduce the way in which matrices are added or multiplied, i ...

MATRICES AND VECTOR SPACES
The matricesA,BandCare given by A= ( 2 − 1 31 ) , B= ( 10 0 − 2 ) , C= ( − 21 − 11 ) . Find the matr ...

8.4 BASIC MATRIX ALGEBRA except for thejth, which equals unity, then we find Aej=      A 11 A 12 ... A 1 N A 21 A 22 ... A ...

MATRICES AND VECTOR SPACES These are clearly not the same, sincePis anM×Mmatrix whilstQis an N×Nmatrix. Thus, particular care mu ...

8.5 FUNCTIONS OF MATRICES TheidentitymatrixIhas the property AI=IA=A. It is clear that, in order for the above products to be de ...

MATRICES AND VECTOR SPACES
Find the transpose of the matrix A= ( 312 041 ) . By interchanging the rows and columns ofAwe immedi ...

8.7 THE COMPLEX AND HERMITIAN CONJUGATES OF A MATRIX
Find the complex conjugate of the matrix A= ( 123 i 1+i 10 ) . By taking t ...

MATRICES AND VECTOR SPACES the right) bybwe obtain a†b=(a∗ 1 a∗ 2 ···a∗N)      b 1 b 2 .. . bN      = ∑N i=1 a∗ibi, (8 ...

8.9 THE DETERMINANT OF A MATRIX which shows that the trace of a multiple product is invariant under cyclic permutations of the m ...

MATRICES AND VECTOR SPACES determinant defined by (8.45) and their corresponding cofactors, we write|A|as the Laplace expansion ...

8.9 THE DETERMINANT OF A MATRIX
Suppose the rows of a real 3 × 3 matrixAare interpreted as the components in a given basis of t ...

MATRICES AND VECTOR SPACES It follows thatanytheorem established for the rows ofAwill apply to the columns as well, and vice ver ...

8.10 THE INVERSE OF A MATRIX
Evaluate the determinant |A|= ∣ ∣ ∣∣ ∣ ∣∣ 1023 01 − 21 3 − 34 − 2 − 21 − 2 − 1 ∣ ∣ ∣∣ ∣ ∣∣ . Takin ...

MATRICES AND VECTOR SPACES equivalent to saying that if we first multiply a matrix,Bsay, byAand then multiply by the inverseA−^1 ...

8.10 THE INVERSE OF A MATRIX
Find the inverse of the matrix A=   243 1 − 2 − 2 −33 2  . We first determine|A|: |A|=2[−2(2)− ...

MATRICES AND VECTOR SPACES and may be straightforwardly derived. (i) (A−^1 )−^1 =A. (ii) (AT)−^1 =(A−^1 )T. (iii) (A†)−^1 =(A−^1 ...

8.11 THE RANK OF A MATRIX 8.11 The rank of a matrix Therankof a generalM×Nmatrix is an important concept, particularly in the so ...

MATRICES AND VECTOR SPACES may be shown that the rank of a generalM×Nmatrix is equal to the size of the largest square submatrix ...

8.12 SPECIAL TYPES OF SQUARE MATRIX leading diagonal, i.e. only elementsAijwithi=jmay be non-zero. For example, A=   10 0 02 0 ...

MATRICES AND VECTOR SPACES Clearly result (8.63) for diagonal matrices is a special case of this result. Moreover, it may be sho ...