Example: Apply the Gram-Schmidt process to find an orthonormal matrix that diagonalizes
Solution
The characteristic polynomial of the matrix is given by
\(= (\lambda + 2)(\lambda^{2} - 2\lambda - {\rm 8})\)
Solving for , we obtain ,
Note that is the eigenvalue of multiplicity
Next, we find the eigenvectors associated with by solving the homogeneous system
Writing the system above in an augmented matrix and applying elementary row operations, we have
which implies
Choosing we get
Thus, basis for the solution space of the linear system consists of the eigenvectors
Since and are not orthogonal, we use the Gram-Schmidt process to obtain an orthonormal basis.
Let
The set is orthogonal.
Normalizing , we obtain
Now, we find the basis for the eigenvector associated with by solving the homogeneous system
Writing the homogeneous system above in augmented matrix and applying elementary row operations
If we choose
Hence a basis for the solution space consists of the eigenvector
Normalizing we have
Therefore, the orthonormal matrix that diagonalizes is
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Matrix Orthonormal Diagonalization
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