Eigenvector Calculations and Verification for y1+y2=5

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SUMMARY

The eigenvector calculations for the equation y1 + y2 = 5 are confirmed as correct, with the characteristic polynomial derived as det(yI - A) = 0, leading to (y - 1)(y - 4) = 0. The eigenvalues of the triangular matrix are 1 and 4, which sum to 5, consistent with the trace of the matrix. The discussion emphasizes the utility of using the trace to simplify eigenvalue calculations without the characteristic polynomial, although the latter is straightforward in this case. Wolfram Alpha (W|A) corroborates these findings for the matrix ((1,0),(3,4)).

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ertagon2
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Can you please confirm my answer?
View attachment 7539
det(yI-A) = 0
(y-1)(y-4)=0
y1+y2=5
 

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Yes, it is correct.

Incidentally, did you already learn about the eigenvalues of triangular matrices?
Did you perhaps already learn about the trace of a matrix (= the sum along the main diagonal)?

This could help you to calculate the required quantity without having to set up the characteristic polynomial. (Of course, that is not much work here, either.)
 
I get the characteristic equation:

$$x^2-(1+4)x+(1\cdot4-0\cdot3)=x^2-5x+4=(x-1)(x-4)=0$$

Hence:

$$\lambda_1+\lambda_2=1+4=5\quad\checkmark$$

W|A confirms our results:

W|A - eigenvalues for ((1,0),(3,4))
 

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