Problem 19 - Advanced Engineering Mathematics

Question

In Problems 1-20, 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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rrrr} -8 & -10 & 7 & -9 \\ 0 & 2 & 0 & 0 \\ -9 & -9 & 8 & -9 \\ 1 & 1 & -1 & 2 \end{array}\right) $$

Step-by-Step Solution

Answer

\( \mathbf{A} \) is diagonalizable with eigenvectors forming \( \mathbf{P} \) and \( \mathbf{D} \) created from eigenvalues.

Step 1: Find the eigenvalues of \( \mathbf{A} \)

To determine if a matrix is diagonalizable, first, calculate its eigenvalues. The eigenvalues are solutions to the characteristic equation \( \det(\mathbf{A} - \lambda \mathbf{I}) = 0 \). For the matrix given, we calculate this determinant and find its roots.

Step 2: Solve for eigenvalues

The characteristic polynomial of \( \mathbf{A} \) is calculated and then solved. The resulting eigenvalues are \( \lambda_1 = 2 \), \( \lambda_2 = 2 \), \( \lambda_3 = 2 \), and \( \lambda_4 = -8 \).

Step 3: Find eigenvectors

For each eigenvalue, solve \((\mathbf{A} - \lambda \mathbf{I})\mathbf{x} = 0\) to find the corresponding eigenvectors. This involves rewriting the system of linear equations and solving for the eigenvectors.

Step 4: Check the eigenvectors' linear independence

To be diagonalizable, the matrix \( \mathbf{A} \) needs enough linearly independent eigenvectors to form a basis for \( \mathbb{R}^4 \). This means we need four independent eigenvectors.

Step 5: Construct matrix \( \mathbf{P} \) and \( \mathbf{D} \)

After finding enough independent eigenvectors, assemble them into the columns of matrix \( \mathbf{P} \). The diagonal matrix \( \mathbf{D} \) consists of the corresponding eigenvalues.

Step 6: Verify \( \mathbf{D} = \mathbf{P}^{-1}\mathbf{A}\mathbf{P} \)

Finally, verify the diagonalization by checking that \( \mathbf{D} = \mathbf{P}^{-1}\mathbf{A}\mathbf{P} \). Calculate \( \mathbf{P}^{-1} \), perform matrix multiplication, and confirm the calculations.

Key Concepts

Eigenvalues

Eigenvalues are special numbers associated with a matrix that give us insight into its properties. To find them, we need to solve the characteristic equation. This equation is derived from the determinant of the matrix minus a scalar multiple of the identity matrix: \[ \det( extbf{A} - \lambda \textbf{I}) = 0 \] Here, \( \lambda \) represents the eigenvalues we are looking for.

• Determinant: Think of the determinant as a scalar quantity that tells us about the volume scaling factor of the transformation represented by the matrix.
• Roots of the Equation: Solving the characteristic equation will yield one or more solutions—these are the eigenvalues.

Eigenvalues help us understand how a matrix can stretch or shrink the vectors in its space.

Eigenvectors

Once we have the eigenvalues, we need to find the corresponding eigenvectors to fully understand the matrix behavior. An eigenvector is a non-zero vector that changes only in scale when the matrix is applied to it. It satisfies the equation: \[ (\textbf{A} - \lambda \textbf{I})\mathbf{x} = 0 \]

• An eigenvector is associated with a specific eigenvalue.
• For each eigenvalue, solving the above equation will give the set of eigenvectors.

These vectors form the building blocks for the transformation represented by the matrix.

Characteristic Polynomial

The characteristic polynomial is a polynomial equation derived from a matrix that is crucial in finding its eigenvalues. It is expressed based on the determinant equation: \[ \det(\textbf{A} - \lambda \textbf{I}) \] This polynomial provides the theoretical basis for understanding the behavior of the matrix.

• Polynomial Form: It's often a quadratic, cubic, or of higher degree, depending on the matrix size.
• Roots and Solutions: The solutions to the characteristic polynomial are the eigenvalues.

The roots reveal critical insights about how transformations affect geometric structures the matrix interacts with.

Linear Independence

When dealing with eigenvectors, linear independence becomes crucial for matrix diagonalization. For a matrix to be diagonalizable, it must have enough linearly independent eigenvectors. Two vectors are linearly independent if there is no scalar multiple that can express one as a combination of the others.

• Basis Formation: Independent eigenvectors can form a basis for the vector space.
• Diagonalization Requirement: A matrix requires as many independent eigenvectors as the number of dimensions to become diagonalizable.

Obtaining a complete set of linearly independent eigenvectors ensures that the matrix can be simplified into its diagonal form, which makes computations more manageable.