Diagonalizing a 3x3 Matrix

Diagonalize the matrix A where A is given as $\begin{bmatrix} 2 & 0 & 0 \\ 1 & 2 & 1 \\ -1 & 0 & 1 \end{bmatrix}$.

Diagonalization is a process that transforms a matrix into a diagonal form, which is often easier to work with. Diagonalization is possible when there are enough linearly independent eigenvectors to populate the columns of the corresponding eigenvector matrix. This process involves finding the eigenvalues of the matrix, which are the roots of the characteristic polynomial obtained by subtracting lambda times the identity matrix from the original matrix and setting the determinant to zero. The corresponding eigenvectors are found by solving the system resulting from substituting each eigenvalue back into the matrix equation. Once the eigenvectors are found, they are used to form a matrix P. If the matrix is diagonalizable, P allows the transformation of the original matrix A into a diagonal matrix D such that A equals P times D times the inverse of P. This property significantly simplifies matrix powers and other computations. Understanding the conditions under which a matrix is diagonalizable, the role of eigenvalues and eigenvectors, and the methodological process of transforming a matrix into diagonal form are key aspects of linear algebra explored in this problem, offering insights into both theoretical and practical scenarios where diagonal matrices can simplify complex problems.