13.37 DIAGONALIZATION OF QUADRATIC FORMS
We know that for every real symmetric matrix A there exists an orthogonal matrix U such that
where λ1, λ2,…, λn are characteristic roots of A.
Applying the orthogonal transformation X = UY to the quadratic form XTAX, we have
If the rank of A is r, then n-r characteristic roots are zero and so
where λ1, λ2,…, λn 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 ...
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