Question #26f69

2 Answers
Jan 25, 2018

The eigenvalues are #(4.61, -2.61)#

Explanation:

The matrix is

#A=((2,3),(4,0))#

To calculate the Eigenvalues, calculate the determinant

#det(A-Ilambda)=0#

where #lambda# is an eigenvalue

#I# is the unit matrix

#A-Ilambda=((2,3),(4,0))-lambda((1,0),(0,1))=((2-lambda,3),(4,-lambda))#

The determinant is

#|(2-lambda,3),(4,-lambda)|=(2-lambda)(-lambda)-12=0#

#-2lambda+lambda^2-12=0#

The caracteristic polynomial equation is

#lambda^2-2lambda-12=0#

#lambda=(-(-2)+-sqrt((-2)^2-4*(1)*(-12)))/(2)#

#=(2+-sqrt(52))/(2)#

#=1+-sqrt13#

The solutions are

#lambda_1=1+sqrt13=4.61#

#lambda_2=1-sqrt13=-2.61#

Jan 25, 2018

#lambda= 1+sqrt(13)color(white)(88)# and #color(white)(88)lambda=1-sqrt(13)#

Explanation:

#A=[(2,3),(4,0)]#

We are looking for:

#AX=K#

Where A is a #nxxn# matrix and K and X are of the order #nxx1#

K is proportional to X, i.e. #K=lambdaX# where #lambda# is a scaler we don't as yet know.

So we are really looking for:

#AX=lambdaK#

Rearranging:

#AX-lambdaX=0#

Factor:

#(A-lambdaI)X=0#

Where I is the identity matrix, and therefore:

#XI=X#

We know that this can only have a non zero solution, if the determinant of #(A-lambdaI)# is zero.

The above relies on you knowing the theory of linear dependence and independence, and it's role in solving homogeneous systems. I am assuming you would know this before dealing with eigenvalues.

Since we require a non zero solution we need to solve:

#|A-lambdaI|=0#

So we find the determinant:

#A-lambdaI=[(2,3),(4,0)]-lambda[(1,0),(4,0)]=[(2-lambda,3),(4,-lambda)]#

#|[(2-lambda,3),(4,-lambda)]|=-lambda(2-lambda)-4xx3=lambda^2-2lambda-12#

By quadratic formula:

#lambda= 1+sqrt(13)color(white)(88)# and #color(white)(88)lambda=1-sqrt(13)#

So these are the eigenvalues of A.

For further help, watch MIT's OpenCourseWare video.

Introduction to Linear Algebra with Gilbert Strang.

Gilbert Strang is an amazing teacher and makes it so much easier to understand.

This is the link for the video on eigenvalues and eigenvectors.

https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-21-eigenvalues-and-eigenvectors/

Hope it helps.