QrQ QQr

is known as an orthogonal matrix. An orthogonal matrix is a matrix for which its inverse equals its transpose.

Theorem 10.3 For the problem in equation (10.8), where A is a symmetric matrix, the eigenvectors that correspond to distinct eigenvalues are pairwise orthogonal and if put together into a matrix, they form an orthogonal matrix.

Proof

Let qi and q? denote the eigenvectors corresponding to A| and Then Aqi = Aiqi qi Aq, = A.,q^ qi and

A<\2 = X2q2 => q[/4q2 = X2q[ q? Since A is symmetric we have that qf Aq2 = qi Aq,

Then

[f Ai £ Aj. the last relationship above implies that qfq: = q;'qi =o

Theorem 10.4 if an eigenvalue X is repeated r times, there will be r orthogonal vectors corresponding to this root.

Example 10.12

Consider the diagonal matrix

0 0

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