task_eigenvalues.tex

% THIS IS WHERE SYMBOLIC COMPUTATION & RANDOMIZATION HAPPENS
\begin{luacode*}
  symcomp = require "symcomp"
  random = require "random"

  local baseMatrices = 
  {
    "[-2, 1] [12, -3]",
    "[-5, -3] [-3, 3]",
    "[4, -5] [-5, 4]"
  }
  
  local k = random.integer(-5, 5)

  local base = symcomp.matrix(random.oneof(baseMatrices))
  A = symcomp.scalarMul(k, base)

  I = symcomp.identityMatrix(2)

  lambdaI = symcomp.scalarMul("lambda", I)
  AminusLambdaI = symcomp.matrixSub(A, lambdaI)

  det = symcomp.det(AminusLambdaI)
\end{luacode*}

\question
Find all eigenvalues and eigenvectors of the matrix
\begin{equation*}
    A = \scprint{A}.
\end{equation*}

\begin{solution}

Calculate $A-\lambda I_2$:

\begin{equation*}
A-\lambda I_2 = \scprint{A} - \lambda\scprint{I} = \scprint{AminusLambdaI}.
\end{equation*}

Then, calculate $\det(A-\lambda I_2)$.

\begin{equation*}
\det(A-\lambda I_2) = \scprint{det}
\end{equation*}

Now, we solve $\det(A-\lambda I_2)=0$.

\scprint{symcomp.printeigenvalues(A, "A")}

\end{solution}