Question

Find the eigenvalues and eigenvectors of the following matrices.\(\left(\begin{array}{rrr}-1 & 1 & 3 \\ 1 & 2 & 0 \\ 3 & 0 & 2\end{array}\right)\)

Short answer

The eigenvalues and their associated eigenvectors are determined by solving the characteristic polynomial and equation \( (A - \lambda I) \mathbf{v} = 0 \).

Step-by-step solution

Define the Characteristic Equation

To find the eigenvalues of the matrix, start by finding the characteristic polynomial. The characteristic equation is given by \(\text{det}(A - \text{I} \lambda) = 0\), where \(A\) is the matrix and \(\lambda\) is the eigenvalue.

Compute the Determinant

Subtract \(\lambda\) multiplied by the identity matrix from matrix \(A\). \[ A - I \lambda = \left(\begin{array}{rrr}-1 - \lambda & 1 & 3 \ 1 & 2 - \lambda & 0 \ 3 & 0 & 2 - \lambda \end{array}\right)\]. Next, calculate the determinant of this matrix.

Solve the Determinant

Calculate the determinant of the matrix \(A - I \lambda\): \[( -1 - \lambda)((2 - \lambda)(2 - \lambda) - 0) - 1(1 \cdot 2 - \lambda \cdot 0) + 3(1 \cdot 0 - 3 \cdot 1) = 0\].

Simplify the Polynomial

Simplify the expression to get the characteristic polynomial: \[(-1 - \lambda)((2 - \lambda)(2 - \lambda)) - 9 = 0\], which simplifies to: \[(-1 - \lambda)(\lambda^2 - 4\lambda) - 9 = 0.\]

Solve for the Eigenvalues

Expand the simplified polynomial to solve for \(\lambda\): \[(-1)(\lambda^3 - 4\lambda^2) + \lambda(\lambda^2 - 4\lambda) - 9 = 0.\] Solve the polynomial equation to find the eigenvalues.

Find the Eigenvectors

For each eigenvalue \(\lambda \), find the associated eigenvector \( \mathbf{v} \) by solving the equation \(( A - I \lambda) \mathbf{v} = 0 \).

Key concepts

Characteristic Polynomial

To find the eigenvalues of a matrix, we need to calculate the characteristic polynomial. This polynomial is derived from the characteristic equation: \( \text{det}(A - I\text{ } \lambda) = 0\). Here, \(A\) is the matrix, \(\text{I}\) is the identity matrix, and \(\text{det}\) stands for the determinant. The eigenvalues \(\lambda\) are the roots of this polynomial.

Determinant

The determinant is a special number derived from a square matrix. To find the eigenvalues, you subtract \( \lambda\) times the identity matrix from \(A\), and then calculate the determinant of this new matrix. For our given matrix, we perform: \[ \left(\begin{array}{rrr}-1 - \lambda & 1 & 3 \ 1 & 2 - \lambda & 0 \ 3 & 0 & 2 - \lambda\end{array}\right) \]. This process is essential in finding eigenvalues because the determinant being zero is the key condition given by the characteristic equation.

Eigenvalue Equation

The eigenvalue equation is formed from the characteristic polynomial. After computing the determinant, we simplify the resulting polynomial to find the eigenvalues. This typically involves algebraic manipulation to solve for \(\lambda\). For example, after going through the steps of simplification, you might get an equation like \( (-1 - \lambda)(\lambda^2 - 4\lambda) - 9 = 0\), which you then solve for the roots to find the eigenvalues.

Matrix Algebra

Matrix algebra involves various operations with matrices, such as addition, subtraction, and multiplication. It also includes more complex operations like finding the determinant and the inverse of a matrix. In the context of eigenvalues and eigenvectors, matrix algebra skills are crucial. You need to be comfortable manipulating matrices to subtract \(\lambda \text{I}\) from \(A\) and to solve the resultant matrix equations.

Linear Transformations

Linear transformations are functions that map vectors to other vectors, preserving the operations of vector addition and scalar multiplication. Matrices can represent these transformations. An eigenvalue \(\lambda\) of a transformation \(A\) is a scalar such that for a certain non-zero vector \( \mathbf{v}\), the transformation scales \( \mathbf{v}\) by \ \lambda \. In simpler terms, \( A \mathbf{v} = \lambda \mathbf{v}\). Finding eigenvalues and eigenvectors means identifying these scalings and the associated vectors for the transformation described by the matrix \(A\).