Calculate the eigenvalues of this matrix: [Note- you\'ll probably want to use a

Question

Calculate the eigenvalues of this matrix: [Note- you'll probably want to use a graphing calculator to estimate the roots of the polynomial which defines the eigenvalues. You can use the web version at x Functions. If you select the "integral curves utility" from the main menu, will also be able to plot the integral curves of the associated differential equations. ] A = [1 4 -4 -16] smaller eigenvalue = associated eigenvector = larger eigenvalue = associated, eigenvector = If y' = Ay is a differential equation, how would the solution curves behave? The solution curves would race towards zero and then veer away towards infinity. (Saddle) The solution curves converge to different points. All of the solutions curves would converge towards 0. (Stable node) All of the solution curves would run away from 0. (Unstable node)

Explanation / Answer

A =
[1 4]
[-4 -16]
characteristic eq
|A-I| = 0
A-I =
[1- 4]
[-4 -16-]
|A-I| = 0
(1-)((-16-) + 16=0
^2 + 15 - 16 + 16 = 0
^2 + 15 = 0
= 0, -15
smaller eigen value = -15
larger eigen value = 0
for = -15
A-I =
[16 4]
[-4 1]...........smaller eigen vector
for = 0
A-I =
[1 4]
[-4 -16]...........larger eigen vector

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