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I am simply trying to find the eigenvalues of a symbolic matrix using Matematica, but it does not give me an answer; it just outputs its input.

(u is a parameter)

y0 = (u - 1)^(1/2);
jacob = {{-1, 0}, {y0, -2 y0^2}} // MatrixForm // Eigensystem

Output:

Why is that so? and How can I solve it?

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Try this

y0 = (u - 1)^(1/2);
jacob = {{-1, 0}, {y0, -2 y0^2}} // Eigensystem

Mathematica graphics

MatrixForm got in the way of your computation.


To be able to use MatrixForm and not have it stop your computation, one option is

y0 = (u - 1)^(1/2);
(jacob = {{-1, 0}, {y0, -2 y0^2}}) // MatrixForm

Mathematica graphics

Eigensystem[jacob]

Mathematica graphics


Added verification by hand per comment. Mathematica result is correct.

$$ A= \begin{pmatrix} -1 & 0\\ \sqrt{u-1} & -2\left( u-1\right) \end{pmatrix} $$ Hence $$ \left\vert A-\lambda I\right\vert = \begin{vmatrix} -1-\lambda & 0\\ \sqrt{u-1} & -2\left( u-1\right) -\lambda \end{vmatrix} $$ Therefore $\left\vert A-\lambda I\right\vert =0$ gives \begin{align*} \left( -1-\lambda\right) \left( -2\left( u-1\right) -\lambda\right) & =0\\ 2u-\lambda+2u\lambda+\lambda^{2}-2 & =0\\ \lambda^{2}+\lambda\left( 2u-1\right) -2+2u & =0 \end{align*} Using the quadratic formula $\lambda=\frac{-b}{2a}\pm\frac{1}{2a}\sqrt {b^{2}-4ac}$ the roots are \begin{align*} \lambda & =\frac{-\left( 2u-1\right) }{2}\pm\frac{1}{2}\sqrt{\left( 2u-1\right) ^{2}-4\left( -2+2u\right) }\\ & =\frac{-2u+1}{2}\pm\frac{1}{2}\sqrt{\left( 2u-3\right) ^{2}}\\ & =\frac{-2u+1}{2}\pm\frac{1}{2}\left( 2u-3\right) \end{align*} Hence \begin{align*} \lambda_{1} & =\frac{-2u+1}{2}+\frac{1}{2}\left( 2u-3\right) \\ & =\frac{-2u}{2}+\frac{1}{2}+u-\frac{3}{2}\\ & =-u+\frac{1}{2}+u-\frac{3}{2}\\ & =-1 \end{align*} And \begin{align*} \lambda_{2} & =\frac{-2u+1}{2}-\frac{1}{2}\left( 2u-3\right) \\ & =\frac{-2u}{2}+\frac{1}{2}-u+\frac{3}{2}\\ & =-u+\frac{1}{2}-u+\frac{3}{2}\\ & =-2u+2\\ & =-2\left( u-1\right) \end{align*}

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  • $\begingroup$ it worked; why was MatrixForm causing problem? $\endgroup$
    – Our
    Apr 17, 2020 at 9:32
  • $\begingroup$ By the way, a side question; which ones are eigenvectors and which ones are eigenvalues? $\endgroup$
    – Our
    Apr 17, 2020 at 9:33
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    $\begingroup$ @onurcanbektas the first list is the eigenvalues. Second is the eigenvectors. They also go in that order with each others. $\endgroup$
    – Nasser
    Apr 17, 2020 at 9:35
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    $\begingroup$ −1 is not an eigenvalue I did not check by hand calculation. I assumed Mathematica is correct. Eigenvalues[jacob] also gives {-1, -2 (-1 + u)}. If you give me a minute or 2, will do it by hand and see. It is possible you discovered a bug otherwise. But need to check. $\endgroup$
    – Nasser
    Apr 17, 2020 at 9:51
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    $\begingroup$ @onurcanbektas I just checked also by hand. Mathematica result is correct :) $\endgroup$
    – Nasser
    Apr 17, 2020 at 10:01

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