Problem 20 - 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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rrrr} 4 & 2 & -1 & 4 \\ 0 & 2 & 0 & 0 \\ 1 & 3 & 2 & 1 \\ 0 & 0 & 0 & 2 \end{array}\right) $$

Step-by-Step Solution

Answer

Matrix \( \mathbf{A} \) is diagonalizable with \( \mathbf{P} \) from eigenvectors and \( \mathbf{D} = \begin{pmatrix} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 4 \end{pmatrix} \).

Step 1: Check Eigenvalues

First, we need to find the eigenvalues of matrix \( \mathbf{A} \). Compute the characteristic polynomial by finding the determinant of \( \mathbf{A} - \lambda \mathbf{I} \), where \( \lambda \) is the eigenvalue and \( \mathbf{I} \) is the identity matrix. The matrix becomes:\[\begin{pmatrix} 4-\lambda & 2 & -1 & 4 \ 0 & 2-\lambda & 0 & 0 \ 1 & 3 & 2-\lambda & 1 \ 0 & 0 & 0 & 2-\lambda \end{pmatrix}\]Calculate the determinant and solve the polynomial to find \( \lambda \). The eigenvalues are \( \lambda_1 = 2, \lambda_2 = 2, \lambda_3 = 4 \).

Step 2: Determine Algebraic and Geometric Multiplicities

Next, confirm that \( \mathbf{A} \) is diagonalizable by comparing algebraic and geometric multiplicities of its eigenvalues. The algebraic multiplicity of \( \lambda_1 = 2 \) is 3 and for \( \lambda_3 = 4 \) is 1. Now, find the null space of \( \mathbf{A} - 2\mathbf{I} \) and \( \mathbf{A} - 4\mathbf{I} \) to determine if the geometric multiplicities also match.

Step 3: Find Eigenvectors

Calculate the eigenvectors for each eigenvalue.1. For \( \lambda = 2 \): Solve \((\mathbf{A}-2\mathbf{I})\mathbf{x} = \mathbf{0}\). You will find two linearly independent eigenvectors.2. For \( \lambda = 4 \): Solve \((\mathbf{A}-4\mathbf{I})\mathbf{x} = \mathbf{0}\). You will find one eigenvector.

Step 4: Formulate Matrix P

Construct the matrix \( \mathbf{P} \) using the eigenvectors collected from Step 3 as columns. Ensure that \( \mathbf{P} \) is invertible by verifying that its eigenvectors are linearly independent. If they are, \( \mathbf{P} \) is:\[\mathbf{P} = \begin{pmatrix} v_1 & v_2 & v_3 & v_4 \end{pmatrix}\] where each \( v_i \) is an eigenvector.

Step 5: Construct Diagonal Matrix D

Form the diagonal matrix \( \mathbf{D} \) using the corresponding eigenvalues on the diagonal in the same order as the eigenvectors in \( \mathbf{P} \). Thus, \( \mathbf{D} \) will be:\[ \mathbf{D} = \begin{pmatrix} 2 & 0 & 0 & 0 \ 0 & 2 & 0 & 0 \ 0 & 0 & 2 & 0 \ 0 & 0 & 0 & 4 \end{pmatrix} \]

Step 6: Verify Diagonalization

Verify if \( \mathbf{A} \) is diagonalizable by checking if \( \mathbf{P}^{-1}\mathbf{AP} = \mathbf{D} \). Calculate \( \mathbf{P}^{-1} \) and multiply to confirm the resulting matrix is \( \mathbf{D} \). If the calculations hold true, matrix \( \mathbf{A} \) is diagonalizable.

Key Concepts

Eigenvalues

When working with matrices, understanding eigenvalues is crucial for determining if a matrix can be diagonalized. Eigenvalues are special numbers associated with a matrix that give us insight into its properties. These values are found by solving the characteristic polynomial, which arises from the equation \( (A - \lambda I) \), where \( A \) is your matrix, \( I \) is the identity matrix of the same dimension, and \( \lambda \) represents the eigenvalues. For our example matrix, the eigenvalues are determined to be \( \lambda_1 = 2 \) (with a multiplicity of three) and \( \lambda_3 = 4 \). Each eigenvalue will help us further explore the behavior of the matrix and decide on its diagonalizability.
Finding the eigenvalues is the first step to understanding how the matrix interacts with vectors. Each eigenvalue presents a different condition or transformation that can be applied to the input eigenvector. By identifying these values, you open the doorway to simplifying complex linear transformations described by matrices.

Eigenvectors

Eigenvectors are vectors associated with a given eigenvalue and provide more context on how the matrix acts on certain spaces. They are solutions to the equation \( (A - \lambda I)\mathbf{x} = \mathbf{0} \), where \( \mathbf{x} \) is a vector. For each eigenvalue, we solve this system of equations to find the corresponding eigenvectors.
For the eigenvalue \( \lambda = 2 \), the solution reveals two linearly independent eigenvectors, forming a basis for a subspace where the matrix scales the vectors by the eigenvalue. Similarly, when \( \lambda = 4 \), there's one eigenvector. These eigenvectors help construct the matrix \( \mathbf{P} \), which is used for diagonalization.

• The geometric meaning is that each eigenvector points in a direction where the transformation \( A \) stretches the vector by its eigenvalue.
• These vectors are critical for constructing the change-of-basis matrix (\( \mathbf{P} \)), necessary for turning \( A \) into a diagonal matrix.

Geometric Multiplicity

The geometric multiplicity of an eigenvalue refers to the number of linearly independent eigenvectors associated with it. Essentially, it tells us how many different directions we can transform into with the matrix regarding that eigenvalue.
For example, in our case where \( \lambda = 2 \), the geometric multiplicity is 2 since there are two linearly independent eigenvectors. Meanwhile, for \( \lambda = 4 \), it is 1, as only one linearly independent eigenvector exists for this eigenvalue. The geometric multiplicity can be less than or equal to the algebraic multiplicity.

• Geometric multiplicity is crucial because it needs to match algebraic multiplicity to ensure diagonalizability.
• If all eigenvalues have geometric multiplicity matching their algebraic multiplicity, the matrix is definitely diagonalizable.

Algebraic Multiplicity

Algebraic multiplicity measures how many times an eigenvalue appears as a root of the characteristic polynomial. In simpler terms, it retells how often an eigenvalue is counted in the matrix.
This number is indicated during the solution of the characteristic equation. In our example, \( \lambda = 2 \) was found to have an algebraic multiplicity of 3. Meanwhile, \( \lambda = 4 \) appears once. These values indicate the spectrum of the matrix's transformation behavior.

• The importance of the algebraic multiplicity is linked to ensuring the matrix is diagonalizable, alongside checking geometric multiplicities.
• Algebraic multiplicity can be seen crucial in understanding potential simplifications to a matrix.

By ensuring that both the algebraic multiplicity and geometric multiplicity match up properly, we can confirm that the matrix can be successfully diagonalized.