Chapter 9

Chapter 9 – last updated on 10.17.17

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I was asked for reference material for tensor math. The extent to which I want you to know tensor math is to the extent required for understanding Chapter 10 (please refer to my already posted slides on our website).

Useful things to know in preparation for chapter 10.:
1) transpose a matrix (should already know): http://www.mathwords.com/t/transpose_of_a_matrix.htm
2) take an inverse of a matrix (should already know; useful for solving equations with vectors/matrices: i.e., slide 39 of chapter 10: http://www.mathwords.com/i/inverse_of_a_matrix.htm
3) multiply permutations of vectors and matrices (should already know, i.e., v*v, v*m, m*v, m*m): https://www.varsitytutors.com/hotmath/hotmath_help/topics/multiplying-vector-by-a-matrix

and

https://www.khanacademy.org/math/precalculus/precalc-matrices/multiplying-matrices-by-matrices/a/multiplying-matrices

4) matrix diagonalization: A = PDinv(P); D = a matrix of the eigenvalues along the diagonal = vector of eigenvalues*identity matrix); you should know whether the vector of eigenvalues is a column or row vector based on our previous discussion of how to multiply 2 tensors together… P = a matrix of eigenvectors as columns. Why is PDinv(P) useful? You can use it to solve for unknowns. For example, AX=Y or PDinv(P)X=Y. You can use these relationships to further understand slide 39 of chapter 10. Slide 39 shows capital lambda with 2 lines under it. This is the same as D discussed above. Please note that in tensor math, 1 line under a variable is a vector (1st order tensor) and 2 lines under a vector is a matrix (2nd order tensor).
5) Calculating eigenvalues and vectors from a matrix A will also be useful.
v(A-lambdaI)=0 where lambda = eigenvalues and I = identity matrix. Note that the 0 is not necessarily a scalar. Note: A = matrix; Lambda*I = D = a matrix). is v a vector? what is 0 in terms of a scalar or a vector? Can 0 be a scalar?

(http://mathworld.wolfram.com/MatrixDiagonalization.html)

 

5) use of an identity matrix (related to diagonalization/inversing matrices: i.e., inv(A)A = I where I = identity matrix and A*inv(A)=I

 

I think if you know how to do these things above you will be poised for chapter 10.