Section 4. Matrix Methods
The LinearAlgebra package is introduced. Matrix and vector manipulation exercises provide practice via eigenvalue and eigenvector calculations. The matrix exponential is a key tool for solving constant coefficient linear systems of differential equations.
0. Pull down the Help menu, choose New Users/Full Tour, then click on the item nine: Linear Algebra and work through the examples in that worksheet.
Use Maple to do the following. Start by loading the LinearAlgebra package via with(LinearAlgebra) .
1. Enter the matrix
using the three methods described below.
a. Using Matrix: Matrix( [ [1,2,3], [4,5,6], [7,8,9] ] )
b. In terms of its columns using the < > notation: <<1,4,7>|<2,5,8>|<3,6,9>>
c. In terms of its rows using the < > notation: <<1|2|3>,<4|5|6>,<7|8|9>>
2. Continuing 1. Enter the matrix in Exercise 1 in terms of its columns using the < > notation. Name it A.
a. Find the determinant and the characteristic polynomial of A.
b. Enter the vector v = <2,3,5> (use the entry v := <2,3,5> ). It will appear as a column vector. Calculate the vector w = Av using the entry w := A.v. It will also be a column vector.
c. Calculate the dot product of v and w using v.w. (Both are column vectors, the product will be a scalar.) Verify that the entry w.v yields the same scalar as does Transpose(v).Transpose(w) and Transpose(w).Transpose(v).
d. Calculate the product Transpose(v).w. This will be the same scalar as in part c conforming to the usual convention for such products. Now enter v.Transpose(w). The output is a 3 x 3 rank 1 matrix (verify by entering Rank(%)).
e. Convert w into a (column) matrix W using W := convert(w,Matrix). Convert v into a (column) matrix V using V := convert(v,Matrix). Now form the products V.W and W.V. Comment on the output.
f. Form the product Transpose(V).W. Compare the Matrix output to the output for Transpose(v).w above.
g. Form the product V.Transpose(W). Compare the Matrix output to the output for v.Transpose(w) above.
Key facts about the "period" product used for Matrices and Vectors: (all properties assume dimensions are compatible)
Matrix times Matrix outputs Matrix.
Matrix times Vector outputs Vector.
Vector times Vector outputs scalar when both of them are columns or both are rows.
Column Vector times Row Vector (in this order) outputs Matrix.
Row Vector times Column Vector (in this order) outputs scalar.
3.* Define the three column vectors b1, b2, b3 as follows
b1,b2,b3 := <0,1,-1>,<1,1,0>,<-1,0,1>;
a. Define the matrix B having b1, b2, b3 as its columns with the entry B := <b1|b2|,b3>.
b. Find the characteristic polynomial of B and factor it with factor(%).
c. Find B's eigenvalues with Eigenvalues(B).
d. Find B's eigenvectors using Eigenvectors(B). Name the input lambda,V.
e. Calculate MatrixInverse(V).B.V.
4.* Continuing 3. Obtain the solution to v' = Bv satisfying v(0) = <1,2,3>.
a. Do if first using dsolve applied to the system of linear differential equations defined by v' = Bv with the appropriate initial conditions.
b. Do it second by making a fundamental matrix solution X(t) defined as the matrix with the eigenvector solutions in the columns then computing
c. To it third by using the Matrix Exponential, Exp_At. Once you have it, the solution is
d. Check the solution in each case.