Section 8.4 Eigenbases. Diagonalization.
Quadratic Forms

Similarity Transformation
(OverHat[A])=(P^(-1))AP

A={{6,-3},{4,-1}}

P={{1,3},{1,4}}

MatrixForm[P]

Eigensystem[A]

AS=Inverse[P].A.P;MatrixForm[AS]

Inverse[P].{1,1}

Inverse[P].{3,4}

Diagonalization: D = (X^(-1))AX

A={{7.3,0.2,-3.7},{-11.5,1.0,5.5},{17.7,1.8,-9.3}}

MatrixForm[A]

Eigenvalues[A]

X=Transpose[Eigenvectors[A]]

Xi=Inverse[X]

X.Xi

MatrixForm[Xi.A.X]

Threshold[%]//MatrixForm

x={{x1},{x2}}

Transpose[x]//MatrixForm

A={{3,4},{6,2}}

Transpose[x].A.x

Expand[%]

A={{17,-15},{-15,17}}

Eigensystem[A]

{{32,2},{{-1,1},{1,1}}}

Eigenvectors[A]

Normalize[{{-1,1},{1,1}},Norm]