Orthogonal Transformation of a Symmetric Matrix to Diagonal Form

ORTHOGONAL TRANSFORMATION OF A SYMMETRIC MATRIX TO DIAGONAL FORM

An n×n matrix A is diagonalizable if and only if it has n linearly independent eigenvectors. Moreover, the matrix P with these eigenvectors as columns is a diagonalizing matrix for A. (ie) P^‒1AP is diagonal.

An orthogonal set of vectors is called orthonormal if || v || = 1 for each vector V in the set and that any orthogonal set { v₁, v₂, v₃... v_k } can be orthonormal set normalized, that is converted into an orthonormal set { v₁/||v₁||, v₂/||v₂||, …. v_k/||v_k|| }. If a matrix A has n orthogonal eigenvectors they can be taken to be orthonormal. The corresponding diagonalizing matrix P has orthonormal columns and such matrices are very easy to invert.

The following conditions are equivalent for an n×n matrix P.

(i) P is invertible and P^‒1 = p^T

(ii) The rows of P are orthonormal.

(iii) The columns of P are orthonormal.

Definition: Orthogonal matrix.

An n×n matrix P is called an orthogonal matrix if it satisfies one and hence all the above three conditions.

Note: If P and Q are orthogonal matrices, then PQ is also orthogonal as is p^‒1= p^T.

Definition: Orthogonally diagonalizable matrices

An n×n matrix A is said to be orthogonally diagonalizable when an orthogonal matrix P can be found such that P^‒1AP−P^TAP is diagonalizable.

Note:

1. The following conditions are equivalent for an n×n matrix A.

(i) A has an orthonormal set of n eigenvectors.

(ii) A is orthogonally diagonalizable.

(iii) A is symmetric.

This is called principal axes theorem (or) real spectral theorem.

2. A set of orthonormal eigenvectors of a symmetric matrix A is called a set of principal axes for A.

3. The set of distinct eigenvalues is called the spectrum of the matrix.

4. If A is an n×n symmetric matrix, then (Ax) y = x (Ay) for all columns x and y in R³.

5. If A is a symmetric matrix, then eigenvectors of A corresponding to distinct eigenvalues are orthogonal.

6. If A is an n×n matrix with n real eigenvalues, an orthogonal matrix P exists such that P^TAP is an upper triangular matrix.

7. If A is an n×n matrix with real eigenvalues λ₁, λ₂, λ₃... λ_n (possibly not all distinct) then |A|= λ₁ λ₂ λ₃... λ_n and trace of A = λ₁ + λ₂ + λ₃ + ... + λ_n