The Chemistry Maths Book, Second Edition

510 Chapter 18Matrices and linear transformations But two matrices need not commute even if they are both square. EXAMPLE 18.10N ...

18.3 Matrix algebra 511 where and A1= 1 h 22 π where his Planck’s constant. The commutation properties of these matrices are [S ...

512 Chapter 18Matrices and linear transformations EXAMPLE 18.12Multiplication by a unit matrix 0 Exercise 43 The product is a ze ...

18.4 The inverse matrix 513 and the trace is (18.40) and this is identical to the sum in (18.39). The transpose of a matrix prod ...

514 Chapter 18Matrices and linear transformations For order 2 the pair of inverse matrices is (ifad 1 − 1 bc 1 ≠ 10 ) (18.43) EX ...

18.4 The inverse matrix 515 2.Transpose the matrix of cofactors: The matrix™is called the adjointofA. 3.Divide the adjoint matri ...

516 Chapter 18Matrices and linear transformations The determinant of this matrix is shown in Example 17.4 to have valueA 1 = 16 ...

18.5 Linear transformations 517 (18.47) that represents the transformation of the vector with components(x 1 , x 2 , =, x n )int ...

518 Chapter 18Matrices and linear transformations (c)Cchanges the sign of the ycoordinate, and represents reflection in the x-ax ...

18.5 Linear transformations 519 EXAMPLE 18.19Rotate the square whose corners have positions in the xy-plane (x, y) 1 = 1 (2, 1), ...

520 Chapter 18Matrices and linear transformations so that Afollowed by Bis equivalent to the single transformation whose matrix ...

18.6 Orthogonal matrices and orthogonal transformations 521 A nonsingular transformation Afollowed by its inverse transformation ...

522 Chapter 18Matrices and linear transformations where (18.58) The transpose of Ais (18.59) where, for example, the row vectora ...

18.6 Orthogonal matrices and orthogonal transformations 523 and therefore,A T 1 = 1 A − 1 . It also follows that the determinant ...

524 Chapter 18Matrices and linear transformations (iii) The determinant of Ais 0 Exercise 61 Orthogonal transformations An ortho ...

18.7 Symmetry operations 525 (ii) A plane of symmetry (Oyz)that bisects the bond angle. Reflection in the plane again results in ...

526 Chapter 18Matrices and linear transformations Other symmetry operations (reflections) are possible, but are equivalent to th ...

18.7 Symmetry operations 527 Axioms of group theory A set of elements{E, P, Q, R, =}forms a group if the following conditions ar ...

528 Chapter 18Matrices and linear transformations represented by the number + 1 , clearly satisfies the group multiplication tab ...

18.8 Exercises 529 representations’) are distinct and independent. In the present case, all the possible representations of the ...