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matrix - Eigenvector Anomaly



I'm trying to compute the eigenvectors for:


$$ M = \left( \begin{array}{ccc} 1 & 4 \\ 4 & 100 \end{array} \right) $$


Both myself and Mathematica report the eigenvalues as:


$$ \lambda_1 = \frac{1}{2} \left(101+\sqrt{9865}\right) \approx 100.161 \\ \lambda_2 = \frac{1}{2} \left(101-\sqrt{9865}\right) \approx 0.838647 $$


But when I ask for eigenvectors, the answer changes depending on whether I input the numbers as integers or floating-point numbers.


When I ask for Eigenvectors[{{1, 4}, {4, 100}}] I get:


$$ v_1 = \left(\frac{1}{8} \left(-99+\sqrt{9865}\right), 1\right) \approx (0.0403383, 1) \\ v_2 = \left(\frac{1}{8} \left(-99-\sqrt{9865}\right), 1\right) \approx (-24.7903, 1) $$


When I ask for Eigenvectors[{{1.0, 4.0}, {4.0, 100.0}}] I get:


$$v_1 \approx (0.0403055, 0.999187) \\ v_2 \approx (-0.999187, 0.0403055)$$


When I calculate by hand, I get a solution which matches the first query.



So, am I going crazy and overlooking some important maths (not unlikely, very tired...) or is this a bug?


Possible related issues?:




Answer



Eigenvectors for inexact arguments are normalized:


Eigenvectors[{{1, 4}, {4, 100}}]
% // N
Normalize /@ %% // N
Eigenvectors[{{1.0, 4.0}, {4.0, 100.0}}]


(*
{{1/8 (-99+Sqrt[9865]),1},{1/8 (-99-Sqrt[9865]),1}}

{{0.0403383,1.},{-24.7903,1.}}

{{0.0403055,0.999187},{-0.999187,0.0403055}}

{{0.0403055,0.999187},{-0.999187,0.0403055}}
*)

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