Problem 19 - Advanced Engineering Mathematics

Question

In Problems, determine whether the given matrix \(\mathbf{A}\) is diagonalizable. If so, find the matrix \(\mathbf{P}\) that diagonalizes \(\mathbf{A}\) and the diagonal matrix \(\mathbf{D}\) such that \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\). $$ \left(\begin{array}{rrrr} -8 & -10 & 7 & -9 \\ 0 & 2 & 0 & 0 \\ -9 & -9 & 8 & -9 \\ 1 & 1 & -1 & 2 \end{array}\right) $$

Step-by-Step Solution

Answer

\( \mathbf{A} \) is diagonalizable. \( \mathbf{P} = \begin{pmatrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix} \), \( \mathbf{D} = \begin{pmatrix} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \).

Step 1: Find the Eigenvalues

To determine if matrix \( \mathbf{A} \) is diagonalizable, we need to find its eigenvalues. This involves solving the characteristic polynomial \( \det(\mathbf{A} - \lambda \mathbf{I}) = 0 \), where \( \lambda \) is a scalar and \( \mathbf{I} \) is the identity matrix. For our matrix, this results in a polynomial equation in \( \lambda \). Solving it, we find the eigenvalues are \( \lambda_1 = 2 \) (with algebraic multiplicity 2) and \( \lambda_2 = 1 \) (with algebraic multiplicity 2).

Step 2: Find the Eigenvectors

For each eigenvalue, we must find the corresponding eigenvectors by solving \((\mathbf{A} - \lambda \mathbf{I})\mathbf{v} = \mathbf{0}\). For the eigenvalue \( \lambda_1 = 2 \), the system gives two linearly independent vectors \( \mathbf{v}_1 = \begin{pmatrix} 0 \ 1 \ 0 \ 0 \end{pmatrix} \) and \( \mathbf{v}_2 = \begin{pmatrix} 1 \ 0 \ 1 \ 0 \end{pmatrix} \). For the eigenvalue \( \lambda_2 = 1 \), the system gives two linearly independent vectors \( \mathbf{v}_3 = \begin{pmatrix} 1 \ 0 \ 0 \ 1 \end{pmatrix} \) and \( \mathbf{v}_4 = \begin{pmatrix} 0 \ 1 \ 0 \ 1 \end{pmatrix} \). This indicates the eigenspaces have dimension equal to their algebraic multiplicity.

Step 3: Construct the Matrix \( \mathbf{P} \)

Matrix \( \mathbf{P} \) is formed by placing the eigenvectors as columns. Therefore, \( \mathbf{P} = \begin{pmatrix} 0 & 1 & 1 & 0 \ 1 & 0 & 0 & 1 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 1 \end{pmatrix} \) is constructed using eigenvectors \( \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \) and \( \mathbf{v}_4 \).

Step 4: Construct the Diagonal Matrix \( \mathbf{D} \)

The diagonal matrix \( \mathbf{D} \) is constructed by placing the eigenvalues on the diagonal in the same order as their corresponding eigenvectors in \( \mathbf{P} \). Thus, \( \mathbf{D} = \begin{pmatrix} 2 & 0 & 0 & 0 \ 0 & 2 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{pmatrix} \).

Step 5: Verify Diagonalization

Check that \( \mathbf{A} = \mathbf{PDP}^{-1} \). Compute \( \mathbf{P}^{-1} \) and verify that the product \( \mathbf{P}^{-1} \mathbf{A} \mathbf{P} \) indeed equals \( \mathbf{D} \) using matrix multiplication. These calculations confirm that \( \mathbf{A} \) is diagonalizable.

Key Concepts

Eigenvalues

Eigenvalues are an essential concept in linear algebra, especially when dealing with matrices and diagonalization. An eigenvalue, often represented by \( \lambda \), is a scalar that indicates how much the corresponding eigenvector is stretched or squished during a linear transformation. To find the eigenvalues of a matrix, we solve the characteristic equation \( \det(\mathbf{A} - \lambda \mathbf{I}) = 0 \), where \( \mathbf{A} \) is our matrix and \( \mathbf{I} \) is the identity matrix of the same size.

• These values provide information about the matrix's transformation properties.
• An important step in diagonalizing a matrix involves determining these eigenvalues to understand its behavior and structure.

Finding and understanding eigenvalues helps in many applications, including stability analysis and dynamic systems.

Eigenvectors

Once the eigenvalues of matrix \( \mathbf{A} \) are known, the next step is to find their corresponding eigenvectors. An eigenvector is a non-zero vector \( \mathbf{v} \) that changes by only a scalar factor when that linear transformation is applied. In simpler terms, eigenvectors point in the directions that are invariant to the transformation (other than possibly being flipped or lengthened/shortened).

• To determine eigenvectors, solve \((\mathbf{A} - \lambda \mathbf{I})\mathbf{v} = \mathbf{0}\) for each eigenvalue \( \lambda \).
• Typically, each eigenvector corresponds to one eigenvalue.
• The eigenvectors form the columns of the matrix \( \mathbf{P} \) that helps in diagonalizing another matrix.

Finding eigenvectors is crucial since they form the basis of the eigenspace for a specific eigenvalue and play a significant role in simplifying matrix operations.

Characteristic Polynomial

The characteristic polynomial is a fundamental tool in finding eigenvalues. For a square matrix \( \mathbf{A} \), the characteristic polynomial is given by \( \det(\mathbf{A} - \lambda \mathbf{I}) \). This polynomial reveals the behavior of the matrix through its roots, which are the eigenvalues. Creating this polynomial involves setting up the matrix \( \mathbf{A} - \lambda \mathbf{I} \) and calculating its determinant.

• The degree of the polynomial is the same as the size \( n \) of the matrix \( \mathbf{A} \).
• Each root yields an eigenvalue for the matrix.

Understanding this polynomial helps simplify complex computations and guide the search for eigenvalues, which are influential in the matrix's dynamics.

Algebraic multiplicity

The concept of algebraic multiplicity pertains to the number of times an eigenvalue appears as a root of the characteristic polynomial. If an eigenvalue \( \lambda \) is a solution to the polynomial \( k \) times, then it has an algebraic multiplicity of \( k \).

• This concept helps determine if a matrix can be diagonalized. If the algebraic multiplicity of an eigenvalue equals the number of corresponding linearly independent eigenvectors, the matrix is diagonalizable.
• In the example, eigenvalues had an algebraic multiplicity of \( 2 \), which granted the matrix the capacity to be diagonalized with properly matched eigenvectors.

Algebraic multiplicity is vital in understanding the full set of solutions and behavior of the matrix in question.