Problem 8 - 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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rr} -2 & -1 \\ 1 & -4 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrix is diagonalizable with matrix \( \mathbf{P} \) and diagonal matrix \( \mathbf{D} \) depending on calculated eigenvectors.

Step 1: Find the Eigenvalues

To determine if the matrix \( \mathbf{A} \) is diagonalizable, first find its eigenvalues. The characteristic equation is obtained from \( \text{det}(\mathbf{A} - \lambda \mathbf{I}) = 0 \). For \( \mathbf{A} = \begin{pmatrix} -2 & -1 \ 1 & -4 \end{pmatrix} \), \( \mathbf{I} \) is the 2x2 identity matrix. Calculate \( \mathbf{A} - \lambda \mathbf{I} = \begin{pmatrix} -2 - \lambda & -1 \ 1 & -4 - \lambda \end{pmatrix} \). The determinant is \( \text{det}(\mathbf{A} - \lambda \mathbf{I}) = (-2-\lambda)(-4-\lambda) + 1 \). Simplifying gives the characteristic polynomial: \( \lambda^2 + 6\lambda + 7 = 0 \). Solve the quadratic equation to find the eigenvalues.

Step 2: Solve the Characteristic Polynomial

Use the quadratic formula \( \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = 6 \), \( c = 7 \). Plugging in these values, the eigenvalues are \( \lambda_1 = \frac{-6 + \sqrt{36 - 28}}{2} = -3 + \sqrt{2} \) and \( \lambda_2 = \frac{-6 - \sqrt{36 - 28}}{2} = -3 - \sqrt{2} \). Since there are two distinct eigenvalues, the matrix is diagonalizable.

Step 3: Find Eigenvectors for Each Eigenvalue

For each eigenvalue, solve \( (\mathbf{A} - \lambda \mathbf{I})\mathbf{x} = \mathbf{0} \) to find the corresponding eigenvector. For \( \lambda_1 = -3 + \sqrt{2} \), substitute into \( \mathbf{A} - \lambda_1 \mathbf{I} \) and solve the resulting system of equations: \( \begin{pmatrix} -2 + 3 - \sqrt{2} & -1 \ 1 & -4 + 3 - \sqrt{2} \end{pmatrix} \). Similarly, solve for \( \lambda_2 \) to find the second eigenvector.

Step 4: Assemble Matrix P and D

The columns of the matrix \( \mathbf{P} \) are the eigenvectors corresponding to \( \lambda_1 \) and \( \lambda_2 \). If the eigenvectors for \( \lambda_1 \) and \( \lambda_2 \) are \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) respectively, then \( \mathbf{P} = \begin{pmatrix} v_1 & v_2 \end{pmatrix} \). The diagonal matrix \( \mathbf{D} \) will have \( \lambda_1 \) and \( \lambda_2 \) on its diagonal: \( \mathbf{D} = \begin{pmatrix} \lambda_1 & 0 \ 0 & \lambda_2 \end{pmatrix} \). Find and specify \( \mathbf{P} \) and \( \mathbf{D} \) based on the eigenvectors computed.

Key Concepts

Eigenvalues

Eigenvalues are special numbers associated with a matrix that provide insight into the matrix's properties. They are scalar values, denoted by \( \lambda \), that result when a matrix \( \mathbf{A} \) transforms a vector \( \mathbf{x} \) in such a way that the vector's direction does not change, although its magnitude might be scaled by the eigenvalue. Mathematically, this is captured by the equation \( \mathbf{A} \mathbf{x} = \lambda \mathbf{x} \).
To find the eigenvalues, one must solve the characteristic equation, which is derived from setting the determinant of \( \mathbf{A} - \lambda \mathbf{I} \) equal to zero. Here, \( \mathbf{I} \) is the identity matrix. For a given matrix, solving this equation will lead to a polynomial equation—the roots of which are the eigenvalues.
For example, when finding the eigenvalues of a 2x2 matrix like \( \begin{pmatrix} -2 & -1 \ 1 & -4 \end{pmatrix} \), the characteristic polynomial is \( \lambda^2 + 6\lambda + 7 = 0 \). Solving this yields two distinct real eigenvalues, demonstrating that the matrix can be diagonalized.

Eigenvectors

Once we have found the eigenvalues of a matrix, the next step is to determine the corresponding eigenvectors. An eigenvector \( \mathbf{x} \) is a nonzero vector which satisfies the equation \( \mathbf{A} \mathbf{x} = \lambda \mathbf{x} \), where \( \lambda \) is an eigenvalue.
To find the eigenvectors, substitute each eigenvalue back into the equation \( (\mathbf{A} - \lambda \mathbf{I})\mathbf{x} = \mathbf{0} \). This gives a system of linear equations that can be solved using simple techniques in linear algebra. Each solution to this system is an eigenvector associated with the given eigenvalue.
For distinct eigenvalues, we typically find one independent eigenvector per eigenvalue. These eigenvectors form the columns of the matrix \( \mathbf{P} \), which is used in diagonalizing the original matrix. Notably, eigenvectors are not unique, as any scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue.

Linear Algebra

Linear algebra is a branch of mathematics dealing with vector spaces and linear mappings between these spaces. It explores concepts like matrices, vectors, and functions that preserve vector addition and scalar multiplication.
Key operations in linear algebra include matrix multiplication, finding determinants, and setting up systems of linear equations. These operations allow us to explore fundamental concepts like base changing, linear transformations, and solving problems related to vector spaces.
In the context of diagonalizing matrices, linear algebra enables the transformation of a matrix into a simpler form, where the primary tools—eigenvalues, and eigenvectors—aid in recasting the matrix into a more manageable diagonal form. Understanding these foundational aspects of linear algebra provides the mathematical language needed to delve into deeper topics like matrix diagonalization and vector space theory.

Matrix Diagonalization

Matrix diagonalization is the process of converting a given square matrix into a diagonal matrix using its eigenvalues and eigenvectors. A matrix is diagonalizable if there exists an invertible matrix \( \mathbf{P} \) and a diagonal matrix \( \mathbf{D} \) such that \( \mathbf{A} = \mathbf{P} \mathbf{D} \mathbf{P}^{-1} \).
This process simplifies matrix computations because working with diagonal matrices is often easier than dealing with the original matrix. Diagonal matrices have non-zero entries only along the main diagonal, while all off-diagonal entries are zero.
Diagonalization involves three main steps:

• Finding the eigenvalues of the matrix.
• Determining the eigenvectors corresponding to each eigenvalue.
• Using the eigenvectors to construct the matrix \( \mathbf{P} \), and placing the eigenvalues on the diagonal of \( \mathbf{D} \).

If you can accomplish all of these, the matrix is successfully diagonalized, opening doors to more advanced linear algebra techniques and solutions. Matrix diagonalization is particularly useful in fields such as systems of differential equations, quantum mechanics, and computer graphics.