Diagonalization of a 3x3 Matrix

Diagonalize the following 3x3 matrix.

Diagonalization is a powerful tool in linear algebra that simplifies many matrix computations by reducing a matrix to its simplest form. The key idea is to express a given square matrix as a product of an invertible matrix, a diagonal matrix, and the inverse of the invertible matrix. When a matrix is diagonalizable, solving systems of linear equations, computing powers of matrices, and finding matrix exponentials become more straightforward.

To start the diagonalization process, you first need to find the eigenvalues of the matrix by solving its characteristic polynomial, which is determined by subtracting lambda times the identity matrix from the original matrix and setting its determinant to zero. Once you have the eigenvalues, the next step is to find the corresponding eigenvectors. An eigenvector, sometimes referred to as a characteristic vector, is a vector that does not change its direction when a linear transformation is applied to it, although it may be scaled by the eigenvalue. This step involves solving a system of equations to determine a basis for each eigenspace associated with each eigenvalue.

If the matrix has n linearly independent eigenvectors (where n is the size of the matrix), then it is diagonalizable. The matrix can then be expressed as a product $PDP^{-1}$, where $P$ is the matrix of eigenvectors and $D$ is the diagonal matrix of eigenvalues.