an example for Schur decomposition

Let

We will find an orthogonal matrix $P$ and an upper triangular matrix $T$ such that $P^{t}AP=T$ applying the proof of Schur’s decomposition.We ’re following the steps below

• We find the eigenvalues of $A$
  The eigenvalues of a matrix are precisely the solutions to the equation

  $$det(\lambda I-A)=0\leftrightarrow {\lambda }^2-\lambda -6=0$$

  
  

  Hence the roots of the quadratic equation (http://planetmath.org/QuadraticFormula) are the eigenvalues${\lambda }_1=-2,{\lambda }_2=3$

• We find the eigenvectors
  For each eigenvalue ${\lambda }_i$, solving the system

  $$(A-{\lambda }_{i}I)X_i=0$$

  So we have thatfor ${\lambda }_1=-2$

  
  

  Analogously for ${\lambda }_2=3$the eigenvector $X_2=(7,-2)$

• We get an orthonormal set of eigenvectors using Gram-Schmidt orthogonalization
  Consider the above two eigenvectors which are linearly independent but are not orthogonal

  $X_1$$=(1,-1)$

  $X_2$$=(7,-2)$

  First we take $w_1=X_1=(1,-1)$. Therefore

  $$w_2=X_2-\frac{w_1\cdot X_2}{{\parallel w_1\parallel }^2}{w}_1$$

  that is,

  $$w_2=(\frac{5}{2},\frac{5}{2})$$

  and finally the orthonormal set is$\{w_1/\parallel w_1\parallel ,w_2/\parallel w_2\parallel \}=\{(\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}}),(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})\}$
  So

  Then