Problem 36, section 7.5:
I have no idea what technology I want to use. That's essentially my
question. What technology are you supposed to use? I only have a TI-83
calculator. Perhaps, more generally, what is the easiest way to find the
determinant of a 6x6 (i think it is) matrix. That's really my question i
guess. Thanks!
You can download Ch 10 from the Guidebook from the following web site:
http://education.ti.com/product/tech/83/guide/83guideus.html
That will tell you how to enter matrices and compute the determinant.
Unfortunately, according to page 3 of that manual, "you can store only
real numbers in TI-83 matrices", so you can't compute the
characteristic polynomial by computing det(x*I-A).
To find the eigenvalue with largest modulus, you'll have to find the
characteristic polynomial of (x*I-A) somehow, find the roots of the
that polynomial (using your calculator), and decide which is largest,
etc. It gets to be pretty messy, and I'll be impressed if you have
enough patience to do this yourself using your TI-83. Here's how
one might do the problem using the computer algebra system I showed
you in class:
> A := MatrixAlgebra(ComplexField(),6)!
[ 1.1, 1.6, 0.6, 0,0,0,
0.82, 0,0,0,0,0,
0, 0.89,0,0,0,0,
0,0,0.81, 0,0,0,
0,0,0,0.53, 0,0,
0,0,0,0,0.29,0];
> f := CharacteristicPolynomial(A);
> S := PolynomialRing(ComplexField());
> f;
0.9999999999999999999999999999*x^6
- 1.099999999999999999999999999*x^5
- 1.311999999999999999999999999*x^4
- 0.4378799999999999999999999999*x^3
> R := Roots(f); R;
[ <0.E-28, 3>, // that means 0 with multiplicity 3
<-0.40397053396925720307615317790063801073 + 0.25751068264945089502870823948610238199*i, 1>,
<-0.40397053396925720307615317790063801073 - 0.25751068264945089502870823948610238199*i, 1>,
<1.9079410679385144061523063557836055371, 1> ]
> lambda := R[4][1];
> Kernel(A-lambda);
Vector space of degree 6, dimension 0 over Real Field
> // nothing, because of ROUND OFF!
> // So, instead we compute an element of the image of f(x)/(x-lambda).
> g := x^3*(x-R[2][1])*(x-R[3][1]);
> g;
x^5 + (0.80794106793851440615230635580127602146 + 0.E-39*i)*x^4 + (0.22950394399399299805147780515909767437 - 3.6734198462218046180000000000000000000 E-40*i)*x^3
> gA := Evaluate(g,A); // g(A)
> v := RMatrixSpace(ComplexField(),6,1)![1,1,0,1,0,1];
> w := gA*v; w; // w has the property that (A-lambda)*w = f(A)*v = 0*v = 0.
[ 50.19374589723473887581627886 + 0.E-39*i]
[ 21.57240196113797254007798730 + 0.E-39*i]
[10.06290921037584139328604946 - 7.238326866179704666000000000 E-40*i]
[4.272121711396122361247122087 - 2.171498059295117855000000000 E-40*i]
[1.186737129929482753099267384 - 1.403529233299195766000000000 E-40*i]
[ 0.1803796634302756929402418790 + 0.E-40*i]
// Now confirm that w is an eigenvector with eigenvalue lambda:
> A*w;
[ 95.76670915100447366349431611 + 0.E-39*i]
[ 41.15887163573248587816934866 + 0.E-39*i]
[ 19.19943774541279556066940869 + 0.E-39*i]
[ 8.150956460404431528561700064 - 5.863044759258627891000000000 E-40*i]
[ 2.264224507039944851460974706 - 1.150893971323966979000000000 E-40*i]
[0.3441537676795499983987875415 - 4.070234775543212890000000000 E-41*i]
> lambda*w;
[ 95.76670915100447366349431611 + 0.E-39*i]
[ 41.15887163573248587816934866 + 0.E-39*i]
[19.19943774541279556066940869 - 1.381030108779668807000000000 E-39*i]
[8.150956460404431528561700064 - 4.143090324476361274000000000 E-40*i]
[2.264224507039944851460974706 - 2.677851064130663871000000000 E-40*i]
[ 0.3441537676795499983987875415 + 0.E-40*i]
> // So it's an eigenvector.