4x4 Matrix Eigenvalues and Eigenvectors

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  • #1
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Homework Statement


I have 4 equations.

3x+6y-5z-t=-8
6x-2y+3z+2t=13
4x-3y-z-3t=-1
5x+6y-3z+4t=-6

I have already solved this matrix using gauss elimination and found that x=1, y=2, z=5, t=-2

Now the next part of the question asks to solve the matrix using eigenvalues and eigenvectors.

Homework Equations


Eigenvalues and Eigenvectors

The Attempt at a Solution



I have worked through to find the eigenvalues and received this equation

0=L^4 - 4(L^3) - 30(L^2) - 162L + 537, where L is lander

from this I found the Eigen values
2.2522
8.7456
-3.4989-3.8757i
-3.4989+3.8757i

Which I am fairly confident are correct.
However I am unsure about how to continue from here and how the corresponding eigenvectors will solve the matrix.

Thanks Very Much
 

Answers and Replies

  • #2
HallsofIvy
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Given that matrix A has eigenvalues [itex]\lambda_1[/itex], [itex]\lambda_2[/itex], [itex]\lambda_3[/itex], and [itex]\lambda_ 4[/itex], with corresponding eigenvectors [itex]v_1[/itex], [itex]v_2[/itex], [itex]v_3[/itex], and [itex]v_4[/itex], form the matrix P having those eigenvectors as columns and diagonal matrix D having the eigenvalues on its diagonal. Then [itex]A= PDP^{-1}[/itex]. The equation Ax= y is the same as [itex]PDP^{-1}x= y[/itex] and then [itex]DP^{-1}x= P^{1}y[/itex], [itex]P^{-1}x= D^{-1}P^{-1}y[/itex], and, finally, [itex]x= PD^{-1}P^{-1}y[/itex]. It is relatively easy to find [itex]P^{-1}[/itex] and [itex]D^{-1}[/itex] is just the diagonal matrix with the reciprocals of the diagonal elements of D on its diagonal.
 
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  • #3
SteamKing
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The Attempt at a Solution



I have worked through to find the eigenvalues and received this equation

0=L^4 - 4(L^3) - 30(L^2) - 162L + 537, where L is lander

Generally, the Greek letter lambda, λ, (not "lander") is used for the unknown eigenvalues, thus the characteristic equation is written:

λ4 - 4λ3 - 30λ2 - 162λ + 537 = 0

from this I found the Eigen values
2.2522
8.7456
-3.4989-3.8757i
-3.4989+3.8757i

Which I am fairly confident are correct.

They are.
 
  • #4
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Generally, the Greek letter lambda, λ, (not "lander") is used for the unknown eigenvalues, thus the characteristic equation is written:

λ4 - 4λ3 - 30λ2 - 162λ + 537 = 0



They are.
thanks, my bad. for some reason everyone I work with calls it lander instead of lambda
 
  • #5
HallsofIvy
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If you are anywhere near Boston, Massachusetts, they may be saying "lamb-der"!
 
  • #6
Ray Vickson
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Homework Statement


I have 4 equations.

3x+6y-5z-t=-8
6x-2y+3z+2t=13
4x-3y-z-3t=-1
5x+6y-3z+4t=-6

I have already solved this matrix using gauss elimination and found that x=1, y=2, z=5, t=-2

Now the next part of the question asks to solve the matrix using eigenvalues and eigenvectors.

Homework Equations


Eigenvalues and Eigenvectors

The Attempt at a Solution



I have worked through to find the eigenvalues and received this equation

0=L^4 - 4(L^3) - 30(L^2) - 162L + 537, where L is lander

from this I found the Eigen values
2.2522
8.7456
-3.4989-3.8757i
-3.4989+3.8757i

Which I am fairly confident are correct.
However I am unsure about how to continue from here and how the corresponding eigenvectors will solve the matrix.

Thanks Very Much

Does your question go on to explain what it means when it says "solve the matrix"? There are many things you can do with eigenvalues/eigenvectors; youcan use them to solve numerous, varied types of "problems", but I cannot figure out what problem the question is now asking you to solve.
 

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