Problem 9 - 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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrix is diagonalizable with \(\mathbf{P} = \begin{bmatrix} 1 & 1 \\ i & -i \end{bmatrix}\) and \(\mathbf{D} = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}\).

Step 1: Find Eigenvalues

To determine if the matrix \(\mathbf{A}\) is diagonalizable, first find its eigenvalues by solving the characteristic equation. The characteristic equation is given by \(\det(\mathbf{A} - \lambda \mathbf{I}) = 0\). For matrix \(\mathbf{A}\), this means solving \(\det\left(\begin{bmatrix} 0-\lambda & 1 \ -1 & 0-\lambda \end{bmatrix}\right) = 0\). This results in the equation \(\lambda^2 + 1 = 0\) leading to eigenvalues \(\lambda = i\) and \(\lambda = -i\).

Step 2: Check Diagonalizability

A matrix is diagonalizable if it has \(n\) distinct eigenvalues or it has \(n\) linearly independent eigenvectors. Here, matrix \(\mathbf{A}\) has complex eigenvalues. To be sure if it's diagonalizable, we need to find two linearly independent eigenvectors corresponding to eigenvalues \(i\) and \(-i\).

Step 3: Find Eigenvectors

For \(\lambda = i\), solve \((\mathbf{A} - i\mathbf{I})\mathbf{v} = 0\), giving the system of equations \(-i \cdot x + y = 0\) and \(-x - iy = 0\). Solving these gives \(\mathbf{v_1} = \begin{bmatrix} 1 \ i \end{bmatrix}\). For \(\lambda = -i\), solve \((\mathbf{A} + i\mathbf{I})\mathbf{v} = 0\), giving the system \(i \cdot x + y = 0\) and \(-x + iy = 0\). Solving these gives \(\mathbf{v_2} = \begin{bmatrix} 1 \ -i \end{bmatrix}\).

Step 4: Form Matrix P

Matrix \(\mathbf{P}\) is formed from the eigenvectors as columns. Thus, \(\mathbf{P} = \begin{bmatrix} 1 & 1 \ i & -i \end{bmatrix}\).

Step 5: Diagonal Matrix D

Matrix \(\mathbf{D}\) will contain the eigenvalues on the diagonal. Therefore, \(\mathbf{D} = \begin{bmatrix} i & 0 \ 0 & -i \end{bmatrix}\).

Step 6: Check Result

Verify that \(\mathbf{D} = \mathbf{P}^{-1}\mathbf{A}\mathbf{P}\) holds true to ensure correctness. Calculate \(\mathbf{P}^{-1}\) and verify the matrix calculations.

Key Concepts

Eigenvalues

Eigenvalues are special numbers associated with a matrix that provide insights into its properties. In the context of diagonalization, they play a crucial role. For a matrix to be diagonalizable, it needs to have enough linearly independent eigenvectors, which generally means having distinct eigenvalues.

To find the eigenvalues of a matrix, you solve its characteristic equation, given by \( \det(\mathbf{A} - \lambda \mathbf{I}) = 0 \), where \( \mathbf{I} \) is the identity matrix of the same size as \( \mathbf{A} \). For the matrix \( \mathbf{A} = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \), solving the characteristic equation \( \lambda^2 + 1 = 0 \) results in the complex eigenvalues \( \lambda = i \) and \( \lambda = -i \).

This indicates the matrix has complex roots, which are crucial in determining its diagonalizability.

Eigenvectors

Eigenvectors provide directions that are stretched or compressed by the transformation represented by a matrix. For each eigenvalue of a matrix, there is a corresponding eigenvector, which can be found by solving the equation \((\mathbf{A} - \lambda \mathbf{I}) \mathbf{v} = 0\).

In our example, with eigenvalues \(i\) and \(-i\), the eigenvectors can be found by setting up the system of equations derived from the matrix subtraction. For \(\lambda = i\), solving the equations yields the eigenvector \(\mathbf{v_1} = \begin{bmatrix} 1 \ i \end{bmatrix} \). For \(\lambda = -i\), the eigenvector \(\mathbf{v_2} = \begin{bmatrix} 1 \ -i \end{bmatrix} \) is acquired.

Both of these eigenvectors are crucial for constructing the matrix \(\mathbf{P}\) needed for diagonalization. The matrix \(\mathbf{P}\) is built by taking these eigenvectors as its columns.

Complex Numbers

Complex numbers are composed of a real part and an imaginary part, typically expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. In this exercise, the solutions to our characteristic equation involve complex numbers: \(i\) and \(-i\).

The imaginary unit \(i\) is defined by the property \(i^2 = -1\). Complex eigenvalues, such as the ones found here, frequently occur in certain types of matrices, particularly those representing rotational transformations.

Working with complex numbers involves understanding their properties and operations such as addition, multiplication, and division, each respecting the rule that \(i^2 = -1\). These operations permit the simplification needed when solving systems for eigenvectors or verifying equations.

Matrix Theory

Matrix theory deals with understanding the properties and functions of matrices. Diagonalization, a process central to matrix theory, transforms a given matrix into a diagonal form, making it easier to analyze and compute with.

In diagonalization, a square matrix \(\mathbf{A}\) is expressed as \(\mathbf{A} = \mathbf{PDP}^{-1}\), where \(\mathbf{D}\) is a diagonal matrix with the eigenvalues of \(\mathbf{A}\) on its diagonal, and \(\mathbf{P}\) is a matrix composed of the eigenvectors of \(\mathbf{A}\).

These transformations simplify tasks such as matrix exponentiation or the computation of powers of the matrix. It is vital to find \(\mathbf{P}\) using linearly independent eigenvectors and ensure that \(\mathbf{P}^{-1}\) exists. As seen with our matrix, diagonalization offers a streamlined view of its structure through eigenvalues and eigenvectors.