In-Class Work, Tuesday November 4
Finding eigenvalues and eigenvectors
Math 221 Linear Algebra, Fall 2021
November 2, 2021

T. Ratliff, Wheaton College
ratliff_thomas@wheatoncollege.edu

Example from Socrative questions

Define the matrix A.
As per usual, I’ll use lower case for any variable

a={{1,2},{3,2}};
a//MatrixForm

Form the characteristic polynomial. You can get λ by entering ESC- lower case l - ESC

Det[a - λ*IdentityMatrix[2]]

Let’s find the roots.
Notice the == in the Solve command

Solve[ Det[a - λ*IdentityMatrix[2]]==0]

Find the eigenspace corresponding to λ=-1 by finding the null space of A - (-1)*IdentityMatrix

a1 = a - (-1)*IdentityMatrix[2];
a1//MatrixForm

RowReduce[a1]//MatrixForm

Thus, a basis for the eigenspace corresponding to λ=-1 is x =(({{-1}, {1}}))
We should always check that Ax = λx

We follow the same process to find the eigenspace corresponding to λ=4

a2=a-4*IdentityMatrix[2];
a2//MatrixForm
RowReduce[a2]//MatrixForm

Thus, a basis for the eigenspace corresponding to λ=4 is x=(({{2/3}, {1}}))

The matrix for in-class work

b={{75/100, 15/100,5/100},{15/100,80/100,10/100},{10/100,5/100,85/100}};
b//MatrixForm