1.8.8 Quadratic forms
condition: Write out a quadratic form with the given matrix \(A\). Bring it to canonical form, determine the rank, positive and negative indices depending on the values of the parameter \( a \). For what values of \( a \) is the form positive definite? \[ A=\left(\begin{array}{ccc} 1 & 2 & -1 \\ 2 & a+4 & 2 a-2 \\ -1 & 2 a-2 & 5 a \end{array}\right) \]
Investigation of quadratic forms, their reduction to canonical and normal forms with finding transformation matrices using Lagrange and orthogonal transformation methods. Positive and negative definite quadratic forms, Sylvester's criterion.