Problem 12 - 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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rrr} 1 & 2 & 2 \\ 2 & 3 & -2 \\ -5 & 3 & 8 \end{array}\right) $$

Step-by-Step Solution

Answer

Matrix \( \mathbf{A} \) is diagonalizable. Find eigenvalues and eigenvectors to form matrices \( \mathbf{P} \) and \( \mathbf{D} \).

Step 1: Find the Eigenvalues

To determine if a matrix is diagonalizable, we need to find its eigenvalues. We start by setting the characteristic equation, which is given by \( \det(\mathbf{A} - \lambda \mathbf{I}) = 0 \), where \( \lambda \) is the eigenvalue and \( \mathbf{I} \) is the identity matrix. Plugging in the given matrix \( \mathbf{A} \), we find \( \mathbf{A} - \lambda \mathbf{I} = \left(\begin{array}{ccc} 1-\lambda & 2 & 2 \ 2 & 3-\lambda & -2 \ -5 & 3 & 8-\lambda \end{array}\right) \). The determinant is calculated as a polynomial in \( \lambda \).

Step 2: Solve the Characteristic Polynomial

The characteristic polynomial from the previous step is \( (1-\lambda)( (3-\lambda)(8-\lambda)+6) - 2(2(8-\lambda) + 6) + 2(-6 - 2\times(3-\lambda)) \). After simplifying, you get a cubic polynomial: \( \lambda^3 - 12\lambda^2 + 32\lambda - 24 = 0 \). This polynomial can be factored or solved using methods such as the Rational Root Theorem to find the eigenvalues.

Step 3: Find the Eigenvectors

For each eigenvalue obtained in the previous step, solve \( (\mathbf{A} - \lambda_i \mathbf{I})\mathbf{v}=0 \) to find the eigenvectors. Substitute each eigenvalue back into \( \mathbf{A} - \lambda \mathbf{I} \) and solve the resulting system of linear equations to find the corresponding eigenvectors.

Step 4: Check Linear Independence of Eigenvectors

Ensure that the eigenvectors found in the previous step are linearly independent. If the number of linearly independent eigenvectors equals the size of the matrix (3 in this case), then \( \mathbf{A} \) is diagonalizable.

Step 5: Form Matrix \( \mathbf{P} \) and Diagonal Matrix \( \mathbf{D} \)

Matrix \( \mathbf{P} \) is formed by using the eigenvectors as its columns. The diagonal matrix \( \mathbf{D} \) is created with the corresponding eigenvalues placed along the diagonal. Check if \( \mathbf{A} = \mathbf{PDP}^{-1} \) to ensure correctness.

Key Concepts

Eigenvalues

Eigenvalues play a crucial role in determining if a matrix is diagonalizable. An eigenvalue tells us how much a corresponding eigenvector stretches or shrinks when a linear transformation is applied by a matrix. To find eigenvalues, we first compute the characteristic equation, \( \det(\mathbf{A} - \lambda \mathbf{I}) = 0 \). Here,

• \( \mathbf{A} \) is the matrix in question,
• \( \lambda \) represents the eigenvalues, and
• \( \mathbf{I} \) is the identity matrix.

Next, solve this determinant. The solutions give us the possible eigenvalues of the matrix \( \mathbf{A} \). Eigenvalues can be real or complex numbers, and the number of eigenvalues equals the size of the matrix (including possible multiplicity). Understanding eigenvalues is the first step in exploring if the matrix can be transformed into a diagonal matrix (diagonalization).

Eigenvectors

Once eigenvalues are found, we can determine their corresponding eigenvectors, which provide critical insights into the structure of the matrix. Each eigenvector is associated with an eigenvalue and is found by solving: \( (\mathbf{A} - \lambda_i \mathbf{I})\mathbf{v}=0 \).

• Here, \( \lambda_i \) is an eigenvalue,
• \( \mathbf{v} \) is the eigenvector, and
• \( \mathbf{I} \) is again the identity matrix.

To solve this, plug an eigenvalue into the matrix expression and solve the resulting system of equations, typically using methods like row reduction or Gaussian elimination. Eigenvectors indicate the directions along which a transformation stretches or compresses. These directions are important in diagonalization, as they are used to form the matrix \( \mathbf{P} \) for diagonal transformation, ensuring the diagonal matrix \( \mathbf{D} \) correctly represents the transformation properties of \( \mathbf{A} \).

Linear Independence

Linear independence is a key concept in determining the diagonalizability of a matrix. When checking if a matrix is diagonalizable, it is essential that its eigenvectors are linearly independent. To check for linear independence of a set of vectors, ensure that no vector in the set can be expressed as a linear combination of the others. Mathematically, if \( c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + ... + c_n\mathbf{v}_n = 0 \) implies \( c_1 = c_2 = ... = c_n = 0 \), the vectors \( \mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_n \) are linearly independent.

• If the number of linearly independent eigenvectors equals the size of the matrix, it identifies that a complete set of directions is present for diagonalization.
• In our context, for a 3x3 matrix, we need exactly three linearly independent eigenvectors.

If the eigenvectors are not linearly independent, the matrix cannot be diagonalized using transformation to a diagonal matrix \( \mathbf{D} \) with matrix \( \mathbf{P} \).

Characteristic Polynomial

The characteristic polynomial is a central tool in finding eigenvalues. It is derived from the expression \( \det(\mathbf{A} - \lambda \mathbf{I}) = 0 \), and expands into a polynomial equation in \( \lambda \). This polynomial is crucial because its roots are the eigenvalues of the matrix. For a 3x3 matrix, the characteristic polynomial is usually a cubic polynomial given by:
\( \lambda^3 + a\lambda^2 + b\lambda + c = 0 \).

• This polynomial's degree matches the size of the matrix, leading to up to three eigenvalues.
• Solving the polynomial can involve factoring if possible or numerical methods when roots are not readily apparent.

Determining the correct roots/eigenvalues via the characteristic polynomial is the foundational step to identifying both eigenvalues and eigenvectors, which are essential in the process of diagonalization of the matrix \( \mathbf{A} \).