Engineering Mathematics

13.37 DIAGONALIZATION OF QUADRATIC FORMS

We know that for every real symmetric matrix A there exists an orthogonal matrix U such that

 

U^T AU = diag [λ₁ λ₂ … λ_n],

 

where λ₁, λ₂,…, λ_n are characteristic roots of A.

Applying the orthogonal transformation X = UY to the quadratic form X_TAX, we have

 

 

If the rank of A is r, then n-r characteristic roots are zero and so

 

 

where λ₁, λ₂,…, λ_n are non-zero characteristic roots.

Definition 13.112.    A square matrix B of order n over a field F is said to be congruent to another square matrix A of order ...