Problem 11 - Advanced Engineering Mathematics

Question

In Problems, 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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\). $$ \left(\begin{array}{rrr} 1 & 0 & 1 \\ 0 & -1 & 3 \\ 0 & 0 & 2 \end{array}\right) $$

Step-by-Step Solution

Answer

Yes, \( \mathbf{A} \) is diagonalizable with \( \mathbf{P} = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \) and \( \mathbf{D} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 2 \end{pmatrix} \).

Step 1: Determine the Eigenvalues

To determine if the matrix \( \mathbf{A} \) is diagonalizable, first find its eigenvalues. We start by finding the determinant of \( \mathbf{A} - \lambda \mathbf{I} \), where \( \mathbf{I} \) is the identity matrix and \( \lambda \) represents the eigenvalues.\[ \mathbf{A} - \lambda \mathbf{I} = \begin{pmatrix} 1-\lambda & 0 & 1 \ 0 & -1-\lambda & 3 \ 0 & 0 & 2-\lambda \end{pmatrix} \]The characteristic polynomial is given by:\[ \text{det}(\mathbf{A} - \lambda \mathbf{I}) = (1 - \lambda)((-1-\lambda)(2-\lambda)) = 0 \]Solving, the eigenvalues of \( \mathbf{A} \) are \( \lambda = 1, \lambda = -1, \lambda = 2 \)

Step 2: Find the Eigenvectors

Proceed by finding the eigenvectors for each eigenvalue. For \( \lambda = 1 \):\[ \mathbf{A} - \mathbf{I} \mathbf{x} = \begin{pmatrix} 0 & 0 & 1 \ 0 & -2 & 3 \ 0 & 0 & 1 \end{pmatrix} = 0 \]The corresponding eigenvector \( \mathbf{x}_1 \) can be found by setting \( x_3 = 0 \), resulting in \( x_1 = (1, 0, 0) \).For \( \lambda = -1 \):\[ \mathbf{A} - (-1)\mathbf{I} \mathbf{x} = \begin{pmatrix} 2 & 0 & 1 \ 0 & 0 & 3 \ 0 & 0 & 3 \end{pmatrix} = 0 \]The corresponding eigenvector \( \mathbf{x}_2 \) can be found by setting \( x_1 = 0 \) and gives \( x_2 = (0, 1, 0) \).For \( \lambda = 2 \):\[ \mathbf{A} - 2\mathbf{I} \mathbf{x} = \begin{pmatrix} -1 & 0 & 1 \ 0 & -3 & 3 \ 0 & 0 & 0 \end{pmatrix} \]The corresponding eigenvector \( \mathbf{x}_3 \) can be found by setting \( x_1 = 1 \) and \( x_2 = 0 \), resulting in \( x_3 = (1, 0, 1) \).

Step 3: Construct Matrix P and Diagonal Matrix D

Having found the eigenvectors, form matrix \( \mathbf{P} \) by placing these eigenvectors as columns:\[ \mathbf{P} = \begin{pmatrix} 1 & 0 & 1 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} \]The diagonal matrix \( \mathbf{D} \) will have the eigenvalues on its diagonal:\[ \mathbf{D} = \begin{pmatrix} 1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 2 \end{pmatrix} \]

Step 4: Verify Diagonalization

Finally, verify that \( \mathbf{D} = \mathbf{P}^{-1} \mathbf{A} \mathbf{P} \) holds. Compute:\( \mathbf{P}^{-1} \) using the formula for the inverse of a matrix. Then, calculate \( \mathbf{P}^{-1} \mathbf{A} \mathbf{P} \) to ensure it equals \( \mathbf{D} \). If all calculations hold, the matrix \( \mathbf{A} \) is diagonalizable with the provided \( \mathbf{P} \) and \( \mathbf{D} \).

Key Concepts

Diagonalization of Matrices

Diagonalization is a process where a square matrix \( \mathbf{A} \) is expressed in the form \( \mathbf{P}^{-1} \mathbf{A} \mathbf{P} = \mathbf{D} \). Here, \( \mathbf{D} \) is a diagonal matrix, and \( \mathbf{P} \) is an invertible matrix whose columns are the eigenvectors of \( \mathbf{A} \).
This process simplifies matrix operations like raising a matrix to a power, as raising a diagonal matrix is straightforward—simply raise each of the diagonal entries to the power. This decomposition method is very useful in various applications such as solving systems of differential equations or analyzing complex systems.
For a matrix to be diagonalizable, it must have enough linearly independent eigenvectors to form the matrix \( \mathbf{P} \). In other words, if the matrix is \( n \times n \), it needs \( n \) independent eigenvectors. This directly relates to having distinct eigenvalues, or repeated eigenvalues where each has a full complement of corresponding eigenvectors. Diagonalization not only reduces complexity but also provides a deeper insight into the geometric and algebraic structure of the matrix.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in linear algebra linked with the behavior of linear transformations. An eigenvalue \( \lambda \) of a matrix \( \mathbf{A} \) is a scalar such that there exists a non-zero vector \( \mathbf{x} \) (the eigenvector) where \( \mathbf{A} \mathbf{x} = \lambda \mathbf{x} \).
This equation implies that the transformation of \( \mathbf{x} \) by \( \mathbf{A} \) results in a vector that is a scaled version of \( \mathbf{x} \), scaled by the factor \( \lambda \). To compute the eigenvalues, you set \( \mathbf{A} - \lambda \mathbf{I} = 0 \), where \( \mathbf{I} \) is the identity matrix and solve for \( \lambda \).
The solutions to this equation, typically given by the roots of the characteristic polynomial, are the eigenvalues. Once the eigenvalues are identified, eigenvectors are found by solving \( (\mathbf{A} - \lambda \mathbf{I})\mathbf{x}=0 \) for each \( \lambda \).

Understanding these elements can provide insights into the properties of \( \mathbf{A} \), such as its stability, distinctiveness of solutions, and its invertibility.

Matrix Theory

Matrix theory is a vast field within linear algebra that deals with matrices, which are rectangular arrays of numbers or functions. Matrices represent linear transformations and are crucial in many aspects of science and engineering.
They are used to describe systems of linear equations, perform operations like transformations and rotations, and represent data or find trends. Some essential components of matrix theory include operations such as addition, multiplication, finding the inverse, and determinants.
In applications, matrices can represent networks, data in physical sciences, or discrete models of real-world systems. Thanks to their ability to compactly represent sets of equations or data, they serve as powerful tools for modeling and analyzing complex systems.
Key concepts like eigenvalue decomposition and diagonalization fit into matrix theory by helping to simplify and understand the structure and behavior of these systems.

Characteristic Polynomial

The characteristic polynomial is a key concept in determining the eigenvalues of a matrix. Given a matrix \( \mathbf{A} \), the characteristic polynomial is formed by taking the determinant of \( \mathbf{A} - \lambda \mathbf{I} \), where \( \lambda \) is a scalar and \( \mathbf{I} \) is the identity matrix.
Expressed usually as \( p(\lambda) = \det(\mathbf{A} - \lambda \mathbf{I}) \), this polynomial is used to find possible eigenvalues of the matrix, as its roots correspond to the eigenvalues. Solving this polynomial is a critical step in determining whether a matrix is diagonalizable.
A matrix with a characteristic polynomial that yields distinct eigenvalues will usually be diagonalizable. Even if eigenvalues are repeated, the multiplicity of the eigenvalue in the characteristic polynomial can determine how many independent eigenvectors exist, influencing diagonalizability.
Knowing the characteristic polynomial assists in understanding not only the eigenstructure but also offers insights regarding matrix behavior and its linear transformations.