Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACES8.26 Show that the quadratic surface5x 2 +11y 2 +5z 2 − 10yz +2xz − 10xy =4is an ellipsoid with semi-axes of lengths 2, 1 and 0.5. Find the direction of itslongest axis.8.27 Find the direction of the axis of symmetry of the quadratic surface7x 2 +7y 2 +7z 2 − 20yz − 20xz +20xy =3.8.28 For the following matrices, find the eigenvalues and sufficient of the eigenvectorsto be able to describe the quadratic surfaces associated with them:⎛(a) ⎝ 5 1 −1⎞ ⎛1 5 1 ⎠ , (b) ⎝ 1 2 2⎞ ⎛2 1 2 ⎠ , (c) ⎝ 1 2 1⎞2 4 2 ⎠ .−1 1 52 2 11 2 18.29 This exercise demonstrates the reverse of the usual procedure of diagonalising amatrix.(a) Rearrange the result A ′ = S −1 AS of section 8.16 to express the originalmatrix A in terms of the unitary matrix S and the diagonal matrix A ′ . Henceshow how to construct a matrix A that has given eigenvalues and given(orthogonal) column matrices as its eigenvectors.(b) Find the matrix that has as eigenvectors (1 2 1) T , (1 − 1 1) T and(1 0 − 1) T , with corresponding eigenvalues λ, µ and ν.(c) Try a particular case, say λ =3,µ = −2 andν = 1, and verify by explicitsolution that the matrix so found does have these eigenvalues.8.30 Find an orthogonal transformation that takes the quadratic formQ ≡−x 2 1 − 2x 2 2 − x 2 3 +8x 2 x 3 +6x 1 x 3 +8x 1 x 2into the formµ 1 y1 2 + µ 2 y2 2 − 4y3,2and determine µ 1 and µ 2 (see section 8.17).8.31 One method of determining the nullity (and hence the rank) of an M × N matrixA is as follows.• Write down an augmented transpose of A, by adding on the right an N × Nunit matrix and thus producing an N × (M + N) array B.• Subtract a suitable multiple of the first row of B from each of the other lowerrows so as to make B i1 =0fori>1.• Subtract a suitable multiple of the second row (or the uppermost row thatdoes not start with M zero values) from each of the other lower rows so as tomake B i2 =0fori>2.• Continue in this way until all remaining rows have zeros in the first M places.The number of such rows is equal to the nullity of A, andtheN rightmostentries of these rows are the components of vectors that span the null space.They can be made orthogonal if they are not so already.Use this method to show that the nullity of⎛⎞−1 3 2 73 10 −6 17A = ⎜−1 −2 2 −3⎟⎝ 2 3 −4 4 ⎠4 0 −8 −4312