• Source: Orthogonal diagonalization

In linear algebra, an orthogonal diagonalization of a normal matrix (e.g. a symmetric matrix) is a diagonalization by means of an orthogonal change of coordinates.
The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form q(x) on




R



{\displaystyle \mathbb {R} }

n by means of an orthogonal change of coordinates X = PY.

Step 1: find the symmetric matrix A which represents q and find its characteristic polynomial



Δ
(
t
)
.


{\displaystyle \Delta (t).}


Step 2: find the eigenvalues of A which are the roots of



Δ
(
t
)


{\displaystyle \Delta (t)}

.
Step 3: for each eigenvalue



λ


{\displaystyle \lambda }

of A from step 2, find an orthogonal basis of its eigenspace.
Step 4: normalize all eigenvectors in step 3 which then form an orthonormal basis of




R



{\displaystyle \mathbb {R} }

n.
Step 5: let P be the matrix whose columns are the normalized eigenvectors in step 4.
Then X = PY is the required orthogonal change of coordinates, and the diagonal entries of




P

T


A
P


{\displaystyle P^{T}AP}

will be the eigenvalues




λ

1


,
…
,

λ

n




{\displaystyle \lambda _{1},\dots ,\lambda _{n}}

which correspond to the columns of P.




References



Maxime Bôcher (with E.P.R. DuVal)(1907) Introduction to Higher Algebra, § 45 Reduction of a quadratic form to a sum of squares via HathiTrust

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