Derive for Windows, Version 4.10

Determinants; Eigenvalues/Eigenvectors; Characteristic Polynomial

The det function calculates the de terminant of a (square) matrix. If M is an
n × n matrix, where n is even, the charpoly(M,x) function computes the
characteristic polynomial for M , using x as the variable. In Derive 4 for Windows,
the charpoly(M,x) function actually returns the determinant ,
which is . Thus, for n odd, Derive 4 for Windows computes
the negative of the characteristic polynomial as defined in the textbook. The
function eigenvalues (M,x) calculates the eigen values for M . Finally, the functions
exact_eigenvector (M,λ) and approx_eigenvector (M,λ) give a general
eigenvector of M for eigenvalue λ. Use the former function when the exact value
of λ is known, but the latter when only a close approximation of λ is known.
(Note: The eigenvector functions are in a utility file known as “vector.mth”.
This file can be loaded by clicking on File, then Load, then Utility.)

Most of these functions are illustrated in Figure D.10, where the matrix in
output line #14 has two eigenvalues , λ₁ = −5, having algebraic and geometric
multiplicity 1, and λ₂ = 3, having algebraic multiplicity 3 and geometric
multiplicity 2. Output line #18 displays a general eigenvector for the (exact)
eigenvalue −3 of the matrix in line #14. Notice that the independent variables
in line #18 are denoted “@2” and “@3” since an independent variable “@1”
was already labeled in line #13. Letting “@2” equal 1, and “@3” equal 0, we
obtain the eigenvector [1, 0, 4, −2], while letting “@2” equal 0, and “@3” equal
1, we obtain the eigenvector [0, 1, 2, −2]. These two eigenvectors form a basis
in R⁴ for the eigenspace E₃.

Input	Output
Author,Matrix,4,4,OK,5,Tab,2,(etc.)
Author, Simplify -ex pressions .html">Expression , charpoly (#14,x),Simplify	#15: x4 −4x3 −18x2+108x − 135
Simplify, Factor (with default Rational), Factor	#16: (x +5)(x − 3)3
Author, Expression,eigenvalues(#14,x),Simplify	#17: [x = 3, x = −5]
File, Load, Utility (select: vector.mth),Open	(“vector.mth loaded”)
Author, Expression, exact_eigenvector(#14,3)

Figure D.10: Derive 4 for Windows session: characteristic polynomial;
eigenvalues; eigenvectors

The characteristic polynomial can also be computed directly using determinants.
The n × n identity matrix is created using the function
identity_matrix(n).

If output line #k consists of a square matrix , a basis of eigenvectors for any
eigenvalue λ of the matrix can also be calculated by row reducing the matrix
“λI_n − #k ”, setting each independent variable in turn equal to 1 with all others
equal to 0, and then solving for the dependent variables.

These ope rations are illustrated in Figure D.11 (see next page) for the matrix
in output line #14. linearly independent eigenvectors for the eigenvector λ₂ =
3 are found from the reduced row echelon form matrix for 3I₄−#14, given in
output line #21. Letting the third column variable equal 1 for the matrix in output
line #21, and its fourth column variable equal 0, we obtain, and
letting its third column variable equal 0 and fourth column variable equal 1, we
obtain . (You can easily verify that spans
the same two-dimensional subspace of R⁴ as the set {[1, 0, 4, −2], [0, 1, 2, −2]}
of eigenvectors obtained earlier from the exact_eigenvector function.)

Gram-Schmidt Process

The necessary calculations for the Gram-Schmidt Process can be performed in
Derive 4 for Windows in the manner illustrated in Figure D.12. We begin
with a given set of three linearly independent vectors {[2, 1, 0, −1], [1, 0, 2, −1],
[0, −2, 1, 0]} in R⁴ (output lines #22 through #24), and construct an orthogonal
basis for the span of those vectors. We then produce an orthonormal basis for
the span by dividing each vector in the orthogonal basis by its length. The abs
function calculates the length of a given vector.

Input	Output
Author, Expression, det (x*identity_matrix(4)-#14),Simplify	#19: x4 − 4x3 − 18x2 +108x − 1
Author, Expression, 3*identity_matrix(4)-#14, Simplify
Author, Expression, row_reduce(#20), Simplify

Figure D.11: Derive 4 for Windows session: characteristic polynomial via determinant;
direct calculation of eigenspace

You can easily verify that (output
lines #22, #25, and #26) is an orthogonal set of vectors spanning the same
subspace of R⁴ as {[2, 1, 0, −1], [1, 0, 2, −1], [0, −2, 1, 0]}. An orthonormal basis
for the same subspace is given by the vectors in output lines #27, #28, and
#29.

Input	Output
Author,Vector,4,OK,2,Tab,1, Tab,0,Tab,-1,OK	#22: [2, 1, 0, −1]
Author,Vector,4,OK,1,Tab,0, Tab,2,Tab,-1,OK	#23: [1, 0, 2, −1]
Author,Vector,4,OK,0,Tab,-2, Tab,1,Tab,0,OK	#24: [0, −2, 1, 0]
Author, Expression, #23-(#23.#22)/(#22.#22)#22,Simplify
Author, Expression, #24-(#24.#22)/(#22.#22)#22 -(#24.#25)/(#25.#25)#25,Simplify
Author, Expression,#22/abs(#22),Simplify
Author, Expression,#25/abs(#25),Simplify
Author, Expression,#26/abs(#26),Simplify

Figure D.12: Derive 4 for Windows session: Gram-Schmidt Process;
orthogonal and orthonormal bases