Mathematical Methods for Physics and Engineering : A Comprehensive Guide

8.12 SPECIAL TYPES OF SQUARE MATRIX Suppose thaty=Axis represented in some coordinate system by the matrix equationy=Ax;then〈y|y ...

MATRICES AND VECTOR SPACES Hence〈y|y〉=〈x|x〉, showing that the action of the linear operator represented by a unitary matrix does ...

8.13 EIGENVECTORS AND EIGENVALUES represent the eigenvectorsxofAin our chosen coordinate system. Convention- ally, these column ...

MATRICES AND VECTOR SPACES We also showed that both Hermitian and unitary matrices (or symmetric and orthogonal matrices in the ...

8.13 EIGENVECTORS AND EIGENVALUES writeA†xi=λ∗ixi. From (8.74) and (8.75) we have (λi−λj)(xi)†xj= 0. (8.76) Thus,ifλi=λjthe eig ...

MATRICES AND VECTOR SPACES a result any arbitrary vectorycan be expressed as a linear combination of the eigenvectorsxi: y= ∑N i ...

8.13 EIGENVECTORS AND EIGENVALUES But (xi)†xiis the modulus squared of the non-zero vectorxiand is thus non-zero. Hence λ∗imust ...

MATRICES AND VECTOR SPACES Henceλ∗=−λand soλmust bepure imaginary(orzero). In a similar manner to that used for Hermitian matric ...

8.13 EIGENVECTORS AND EIGENVALUES eigenvector ofAcorresponding to eigenvalueλi. But the eigenvector solutions of (A−λiI)xi= 0 ar ...

MATRICES AND VECTOR SPACES 8.14 Determination of eigenvalues and eigenvectors The next step is to show how the eigenvalues and e ...

8.14 DETERMINATION OF EIGENVALUES AND EIGENVECTORS Expanding out this determinant gives (1−λ)[(1−λ)(− 3 −λ)−(−3)(−3)]+1[(−3)(3)− ...

MATRICES AND VECTOR SPACES
Construct an orthonormal set of eigenvectors for the matrix A=   103 0 − 20 301  . We first dete ...

8.15 CHANGE OF BASIS AND SIMILARITY TRANSFORMATIONS and representxin this basis by the column matrix x=(x 1 x 2 ···xn)T, having ...

MATRICES AND VECTOR SPACES Comparing this with the second equation in (8.93) we find that the components of the linear operatorA ...

8.16 DIAGONALISATION OF MATRICES orthonormal and the transformation matrixSis unitary then 〈e′i|e′j〉= 〈∑ k Skiek ∣ ∣ ∣ ∑ r Srjer ...

MATRICES AND VECTOR SPACES that is,Sij=(xj)i. ThereforeA′is given by (S−^1 AS)ij= ∑ k ∑ l (S−^1 )ikAklSlj = ∑ k ∑ l (S−^1 )ikAkl ...

8.16 DIAGONALISATION OF MATRICES
Diagonalise the matrix A=   103 0 − 20 301  . The matrixAis symmetric and so may be diagon ...

MATRICES AND VECTOR SPACES |expA|. Moreover, by choosing the similarity transformation so that it diagonalisesA,we haveA′=diag(λ ...

8.17 QUADRATIC AND HERMITIAN FORMS i.e.Qis unchanged by considering only the symmetric part ofM. Hence, with no loss of generali ...

MATRICES AND VECTOR SPACES also. Another, rather more general, expression that is also real is theHermitian form H(x)≡x†Ax, (8.1 ...