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

Step-by-Step Solution

Answer

\( \mathbf{A} \) is not diagonalizable.

Step 1: Find the Eigenvalues

Determine eigenvalues of \( \mathbf{A} \). Start by solving the characteristic equation \( \det(\mathbf{A} - \lambda \mathbf{I}) = 0 \). The determinant is given by \( \det\left(\begin{array}{cc} - \lambda & 1 \ -1 & 2 - \lambda \end{array}\right) = \lambda^2 - 2\lambda + 1 = 0 \). Factor this to find \( (\lambda - 1)^2 = 0 \), so the eigenvalue \( \lambda = 1 \) with algebraic multiplicity 2.

Step 2: Find the Eigenvectors

Solve \((\mathbf{A} - \lambda \mathbf{I})\mathbf{v} = \mathbf{0}\) for each eigenvalue. For \( \lambda = 1 \), solve \((\mathbf{A} - \mathbf{I}) = \left(\begin{array}{cc} -1 & 1 \ -1 & 1 \end{array}\right) \mathbf{v} = \mathbf{0}\). Solving this gives \( \mathbf{v} = \left(\begin{array}{c} x \ x \end{array}\right) \text{ for any } x \). The geometric multiplicity is 1, as there is only one linearly independent eigenvector: \( \mathbf{v}_1 = \left(\begin{array}{c} 1 \ 1 \end{array}\right) \).

Step 3: Check Diagonalizability

A matrix is diagonalizable if and only if the geometric multiplicity of each eigenvalue matches its algebraic multiplicity. Here, the eigenvalue \( \lambda = 1 \) has algebraic multiplicity 2 but geometric multiplicity 1. Therefore, \( \mathbf{A} \) is not diagonalizable.

Key Concepts

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in linear algebra, particularly when analyzing matrices. An eigenvalue \( \lambda \) of a matrix \( \mathbf{A} \) is a scalar such that there exists a non-zero vector \( \mathbf{v} \) satisfying the equation \( \mathbf{A} \mathbf{v} = \lambda \mathbf{v} \). Here, \( \mathbf{v} \) is called an eigenvector corresponding to the eigenvalue \( \lambda \). This definition implies that when \( \mathbf{A} \) acts on \( \mathbf{v} \), it simply scales \( \mathbf{v} \) by \( \lambda \) rather than changing its direction. This can simplify many complex matrix operations, particularly those involving repeated application of the matrix.

To find the eigenvalues, one typically solves the characteristic equation related to the matrix, which allows you to identify scalar values that satisfy the eigenvalue condition. Once found, these values help in further exploring the properties of the matrix, such as its diagonalizability.

Linear Algebra

Linear algebra encompasses the branch of mathematics concerning vector spaces and linear mappings between these spaces. This includes the study of lines, planes, and subspaces, but goes much deeper into dimensions beyond three. Among its core operations are addition, scalar multiplication, and finding the dot product between vectors.

It provides essential tools used for solving systems of linear equations. This is done using methods like Gaussian elimination, matrix operations, and vector space inspection.
Linear algebra is not only a theoretical field but is highly applicable in various fields such as physics, computer science, and engineering, where linear models and equations simplify complex problems.

Algebraic and Geometric Multiplicity

Multiplicity is a concept that helps in understanding the behavior of eigenvalues of matrices.

• Algebraic Multiplicity (AM): Refers to the number of times an eigenvalue appears as a root of the characteristic polynomial of a matrix. For instance, if an eigenvalue \( \lambda \) appears twice in the polynomial equation, its algebraic multiplicity is 2.
• Geometric Multiplicity (GM): The number of linearly independent eigenvectors corresponds to a particular eigenvalue. It indicates the dimensionality of the eigenspace associated with that eigenvalue.

Comparing these two types of multiplicities helps determine if a matrix is diagonalizable. A matrix is only diagonalizable if the algebraic multiplicity equals the geometric multiplicity for each of its eigenvalues. If this condition is not met, as in the example matrix where AM is 2 and GM is 1, the matrix cannot be diagonalized.

Characteristic Equation

The characteristic equation of a matrix is essential to finding its eigenvalues. It is derived by setting the determinant of \( \mathbf{A} - \lambda \mathbf{I} \) equal to zero, where \( \mathbf{A} \) is the matrix in question, \( \lambda \) represents eigenvalues and \( \mathbf{I} \) is the identity matrix with the same dimensions as \( \mathbf{A} \). This results in a polynomial equation whose roots are the eigenvalues of \( \mathbf{A} \).

The coefficients and properties of this polynomial give insight into the behavior and characteristics of the matrix. Solving the characteristic equation gives a crucial first step towards understanding a matrix's structure and determining if it can be diagonalized. In the original exercise, the characteristic equation results in a double root, \( (\lambda - 1)^2 = 0 \), yielding repeated eigenvalues and signaling the need to check for diagonalizability with multiplicities.