Diagonalization is a process by which a square matrix ( A ) is transformed into a diagonal matrix ( D ) through a similarity transformation involving an invertible matrix ( P ). The diagonal matrix ( D ) has the eigenvalues of ( A ) on its main diagonal, and ( P ) is composed of the corresponding eigenvectors.
The general formula for diagonalization is given by:
D = P^{1} A P
Here, ( D ) is the diagonal matrix, and ( P ) is the matrix composed of the eigenvectors of ( A ).
1.Find Eigenvalues: Solve the characteristic equation to find the eigenvalues.
2. Find Eigenvectors: For each eigenvalue , find the corresponding eigenvector by solving the system.
3. Form Matrix P: Arrange the eigenvectors as columns to form the matrix ( P ).
4. Form Diagonal Matrix D: The diagonal matrix ( D ) is formed using the eigenvalues on the main diagonal.
It's important to note that not all matrices are diagonalizable. A matrix is diagonalizable if and only if it has ( n ) linearly independent eigenvectors, where ( n ) is the size of the matrix.

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