Question

In each case, decide whether the matrix \(A\) is diagonalizable. If so, find \(P\) such that \(P^{-1} A P\) is diagonal. $$ \text { a. }\left[\begin{array}{lll} 1 & 0 & 0 \\ 1 & 2 & 1 \\ 0 & 0 & 1 \end{array}\right] \quad \text { b. }\left[\begin{array}{rrr} 3 & 0 & 6 \\ 0 & -3 & 0 \\ 5 & 0 & 2 \end{array}\right] $$ c. \(\left[\begin{array}{rrr}3 & 1 & 6 \\ 2 & 1 & 0 \\ -1 & 0 & -3\end{array}\right]\) d. \(\left[\begin{array}{lll}4 & 0 & 0 \\ 0 & 2 & 2 \\ 2 & 3 & 1\end{array}\right]\)

Short answer

Matrices b, c, and d are diagonalizable. Matrix a is not diagonalizable.

Step-by-step solution

Understand the Concept of Diagonalization

A matrix is diagonalizable if there exists a matrix \( P \) such that \( P^{-1} A P \) is a diagonal matrix. This is possible if and only if \( A \) has enough linearly independent eigenvectors to form the columns of \( P \).

Calculate Eigenvalues

To determine diagonalizability, calculate the eigenvalues of each matrix by solving the characteristic equation \( \text{det}(A - \lambda I) = 0 \) for \( \lambda \).

Solve for Eigenvectors for Matrix A

For each eigenvalue found, calculate the corresponding eigenvectors by solving \((A - \lambda I)\mathbf{v} = 0\), where \( \mathbf{v} \) is the eigenvector.

Check Diagonalizability for matrix a

For matrix a, calculate the eigenvalues: \( \lambda = 1, 2 \) each with apparent algebraic multiplicity. Eigenvector calculation shows only 1 unique eigenvector, hence it cannot be diagonalized as it needs three linearly independent eigenvectors.

Check Diagonalizability for matrix b

For matrix b, calculate the eigenvalues: \( \lambda = 3, -3 \). Compute eigenvectors, and find that there are three linearly independent eigenvectors. Thus, matrix b is diagonalizable and \( P \) is formed by these eigenvectors.

Check Diagonalizability for matrix c

Eigenvalues for matrix c include \( \lambda = 3, 1, -3 \). Calculation of eigenvectors shows three linearly independent eigenvectors. Hence, matrix c is diagonalizable, and \( P \) comprises these eigenvectors.

Check Diagonalizability for matrix d

The eigenvalues of matrix d are \( \lambda = 4, 2, 1 \). Solving for eigenvectors indicates three linearly independent eigenvectors, making matrix d diagonalizable. \( P \) comes from these eigenvectors.

Key concepts

Eigenvalues

Eigenvalues are fundamental in determining if a matrix is diagonalizable. These values are derived from the solution to the characteristic equation: \( \text{det}(A - \lambda I) = 0 \). Here, \( A \) is your matrix and \( I \) is the identity matrix of the same size. The parameter \( \lambda \), which satisfies this equation, is considered an eigenvalue of the matrix.

• Eigenvalues represent the scaling factor by which a corresponding eigenvector is stretched or compressed.
• To determine diagonalizability, the number of independent eigenvectors should match the size of the matrix.

If all eigenvalues 'fit' into this framework—meaning each distinct eigenvalue has at least one eigenvector—the matrix likely possesses a complete set of linearly independent eigenvectors. This is a strong indication of diagonalizability.

Eigenvectors

Eigenvectors are specific vectors associated with a matrix that, when transformed by the matrix, only scale by the corresponding eigenvalue and do not change direction. For each eigenvalue found from the characteristic equation, a corresponding eigenvector is determined by solving \((A - \lambda I)\mathbf{v} = 0\), where \(\mathbf{v}\) is the eigenvector.

• Eigenvectors provide the directions in which the matrix acts simply as scaling factors.
• To find them, you solve a homogeneous system which arises from substituting each eigenvalue back into \(A - \lambda I\).

It's important to verify whether these vectors are linearly independent. Only when there are as many independent eigenvectors as the dimension of the matrix, can diagonalization be achieved. Each distinct eigenvalue should manifest at least one independent eigenvector.

Diagonal Matrix Transformation

Diagonal matrix transformation is a process that simplifies the study and use of matrices. By diagonalizing a given matrix, you convert it into a diagonal matrix, where all its non-diagonal elements are zero, and the diagonal elements are the eigenvalues of the matrix. This is achieved through a similarity transformation: \( P^{-1} A P = D \), where \( D \) is a diagonal matrix.

• The matrix \( P \), composed of the eigenvectors, allows the transformation of \( A \) into a diagonal form.
• A diagonal matrix is easier to work with, especially for computations such as exponentials of matrices.

Diagonal matrix transformation is valid if and only if the matrix has enough independent eigenvectors to form \( P \). The diagonalization process not only simplifies mathematical operations but also provides deeper insights into the properties of the original matrix, such as its determinants and powers.