The Chemistry Maths Book, Second Edition

530 Chapter 18Matrices and linear transformations Find the commutator of the following pairs of matrices: 42.The spin ma ...

18.8 Exercises 531 represents an anticlockwise rotation through angle θabout the z-axis. (i) Write down the corresponding linear ...

19 The matrix eigenvalue problem 19.1 Concepts The determination of the eigenvalues and eigenvectors of a square matrix is calle ...

19.1 Concepts 533 or Ax 1 = 1 b (19.3) We saw in Section 17.4 that the equations have a unique solution whendet 1 A 1 ≠ 10 , and ...

534 Chapter 19The matrix eigenvalue problem The matrix equation (19.3) represents the case of inhomogeneous equations, with b 1 ...

19.2 The eigenvalue problem 535 The left side of this equation is called the characteristic (or secular)determinantof the matrix ...

536 Chapter 19The matrix eigenvalue problem with eigenvalues λ 1 1 = 1 − 1 , λ 2 1 = 11 , and λ 3 1 = 12. The corresponding solu ...

19.3 Properties of the eigenvectors 537 (ii) The characteristic equation of the symmetric matrix with nondegenerate eigenvalueλ ...

538 Chapter 19The matrix eigenvalue problem EXAMPLE 19.5Normalization of eigenvectors The eigenvectors xof Example 19.3 are norm ...

19.3 Properties of the eigenvectors 539 so that, whenλ k 1 ≠ 1 λ l , x l T x k 1 = 10 (19.19) and the vectors are orthogonal. EX ...

540 Chapter 19The matrix eigenvalue problem It is always possible to find two linear combinations of the vectors that are orthog ...

α+ 2 β α α− 2 β E 1 E 2 E 3 E 4 E .. ... .. ... .. ... .. ... ... .. .. ... ... .. ... .. .. ... ... .. ... .. ... .. ... .. ... ...

542 Chapter 19The matrix eigenvalue problem The secular equations of the problem are (1) (α 1 − 1 E)c 1 1 + 1 βc 2 1 + βc 4 = 10 ...

19.4 Matrix diagonalization 543 wherec 2 andc 3 are arbitrary. The normalized vectors are also orthogonal; they form an orthonor ...

544 Chapter 19The matrix eigenvalue problem EXAMPLE 19.10The eigenvectors of the matrix (Examples 19.2 and 19.3) are (ignoring t ...

19.4 Matrix diagonalization 545 EXAMPLE 19.11Diagonalization The inverse of the matrix Xof the eigenvectors of Ain Example 19.10 ...

546 Chapter 19The matrix eigenvalue problem It follows from these invariance properties that, for a square matrix A, The trace ...

19.5 Quadratic forms 547 EXAMPLE 19.13Verify that We have and 0 Exercises 29, 30 The general quadratic form in nvariables is (19 ...

548 Chapter 19The matrix eigenvalue problem The canonical form We have seen (equation (19.30)) that a symmetric matrix Ais reduc ...

19.5 Quadratic forms 549 corresponding to eigenvaluesλ 1 1 = 11 andλ 2 1 = 19. The matrix of the eigenvectors is with transpose ...