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Matrix Orthonormal Diagonalization

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Eli
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#1

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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