ECH5842 Chemical Engineering Analysis Fall'97

Homework#3

1(a) Show that any tensor can be represented as:

$${\bf T = D + W}$$

where D is symmetric and W is skew- symmetric. Show that this representation is unique. Also prove that D:W = 0.

1(b) Using the component representation of a tensor, and the operator definitions of the invariants, show that:

$${\rm I}_{\rm A} = {\rm A}_{\rm ii} \qquad \qquad {\rm I I}_{... ...rm ii} {\rm A}_{\rm jj} - {\rm A}_{\rm ij} {\rm A}_{\rm ji} )$$

[Hint: Use an orthonormal basis $\{ \hat{\rm e}_{\rm i} \}$ for the set of arbitrary vectors in the operator definitions]

2 (a) Prove the identity: (ac:bd) = (a.d)(b.c)
(i) What is an obvious eigenvector of the dyadic product ab?
(ii) What does the first scalar invariant of ab correspond to?
(iii) Show that the third scalar invariant of ab vanishes.

2(b) Show that the components of a tensor in two orthonormal bases $\{ {\rm e}_{\rm i}\}$ and $\{ {\rm e}_{\rm i}^\prime \}$, resulting from a rotation, are related by:

$${\rm A}_{\rm ij}^\prime = \alpha_{\rm ki} \alpha_{\rm lj} {\rm A}_{\rm kl}$$

3. For the matrix:

$$A = \left( \begin{array} {cccc} 1 & 1 & 2 & 0 \ 0 & 1 & 3 & 0 \ 0 & 0 & 2 & 2 \ 0 & 0 & 0 & 1 \end{array} \right)$$

(a) show that the eigenvalues are $\lambda_1 = \lambda_2 = \lambda_3 = 1$,$\lambda_4 = 2$, and the eigenvectors:

$$e_1 = \left( \begin{array} {c} 1 \ 0 \ 0 \ 0 \end{arra... ...eft( \begin{array} {c} 5 \ 3 \ 1 \ 0 \end{array} \right)$$

If A were symmetric we would be able to find three orthogonal eigenvectors corresponding to the triple eigenvalue. But here we cannot. However, we can find the so-called generalized eigenvectors e₂ and e₃ such that e₁, e₂, e₃, and e₄ are at least linearly independent. We introduce e₂ and e₃ as follows:

(b) Show that e₁, $\ldots$, e₄ are necessarily linearly independent.

(c) Determine e₂ and e₃ for this example.

4(a) Two matrices A and B are said to be similar, if there exists an invertible matrix C such that A=C^-1BC. Show that similar matrices have the same eigenvalues.

4(b) Show that a symmetric matrix can always be diagonalized by the transformation P^T A P where P is a modal matrix (the matrix whose columns are the eigenvectors of A) of A, with elements corresponding to the eigenvalues of A. Show also that P^T = P^-1.

5. (Jordan canonical form) Any square matrix can be almost diagonalized, more specifically, triangularized, as follows: let Q be a matrix whose columns are the eigenvectors and generalized eigenvectors e_j of A (see problem 3). Since the e_j's are linearly independent, it follows that det $Q \neq 0$,so that Q^-1 exists. If we denote the rows of Q^-1 as r_j, then $r_i \cdot e_j = \delta_{ij}$.The matrix Q^-1 A Q will be in the Jordan canonical form; that is the main diagonal will consist of the $\lambda_j$'s, and the off-diagonal elements will all be zero except for the elements immediately above the repeated $\lambda$'s which may be unity. Verify these claims for the matrix of the problem 3.