Question

Find the eigenvalues of the given matrix. For each eigenvalue, give an eigenvector. $$\left[\begin{array}{ccc}5 & -2 & 3 \\ 0 & 4 & 0 \\ 0 & -1 & 3\end{array}\right]$$

Short answer

Eigenvalues: 5, 4, 3. Eigenvectors: \( \begin{pmatrix} 3 \\ 0 \\ 1 \end{pmatrix} \), \( \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix} \), \( \begin{pmatrix} 3 \\ 0 \\ 2 \end{pmatrix} \).

Step-by-step solution

Define the problem

We want to find the eigenvalues and eigenvectors of the matrix \( A = \begin{pmatrix} 5 & -2 & 3 \ 0 & 4 & 0 \ 0 & -1 & 3 \end{pmatrix} \). Eigenvalues are found by solving the characteristic equation \( \det(A - \lambda I) = 0 \).

Calculate the characteristic equation

Subtract \( \lambda \) times the identity matrix from \( A \), resulting in \( A - \lambda I = \begin{pmatrix} 5-\lambda & -2 & 3 \ 0 & 4-\lambda & 0 \ 0 & -1 & 3-\lambda \end{pmatrix} \). Compute the determinant: \[ \det(A - \lambda I) = (5-\lambda)((4-\lambda)(3-\lambda)) - (-2)(0) + 3(0) = 0 \].

Simplify and solve the characteristic polynomial

Simplify the determinant to get the polynomial equation: \( (5-\lambda)((4-\lambda)(3-\lambda)) = 0 \). Expanding the terms, \( (5-\lambda)(12 - 7\lambda + \lambda^2) = 0 \). Continuing this expansion results in \( \lambda^3 - 12\lambda^2 + 35\lambda - 40 = 0 \).

Find eigenvalues

Solve the cubic equation \( \lambda^3 - 12\lambda^2 + 35\lambda - 40 = 0 \). Trying potential values for \( \lambda \) or using factoring methods, we find eigenvalues: \( \lambda_1 = 5 \), \( \lambda_2 = 4 \), \( \lambda_3 = 3 \).

Find eigenvectors for each eigenvalue

For each eigenvalue, solve \( (A - \lambda I)\mathbf{v} = 0 \) to find eigenvectors.- **For \( \lambda_1 = 5 \):** \( (A - 5I) = \begin{pmatrix} 0 & -2 & 3 \ 0 & -1 & 0 \ 0 & -1 & -2 \end{pmatrix} \). Solve for \( \mathbf{v} \), assume \( x_3 = t \), get \( \mathbf{v} = t\begin{pmatrix} 3 \ 0 \ 1 \end{pmatrix} \). - **For \( \lambda_2 = 4 \):** \( (A - 4I) = \begin{pmatrix} 1 & -2 & 3 \ 0 & 0 & 0 \ 0 & -1 & -1 \end{pmatrix} \). Solve for \( \mathbf{v} \), assume \( x_3 = s \), get \( \mathbf{v} = s\begin{pmatrix} 2 \ 1 \ 0 \end{pmatrix} \). - **For \( \lambda_3 = 3 \):** \( (A - 3I) = \begin{pmatrix} 2 & -2 & 3 \ 0 & 1 & 0 \ 0 & -1 & 0 \end{pmatrix} \). Solve for \( \mathbf{v} \), assume \( x_3 = r \), get \( \mathbf{v} = r\begin{pmatrix} 3 \ 0 \ 2 \end{pmatrix} \).

Verify solutions

Check if the solutions satisfy \( A\mathbf{v} = \lambda\mathbf{v} \) for each pair of eigenvalue and corresponding eigenvector to ensure the calculations are correct.

Key concepts

Eigenvectors

When working with matrices, eigenvectors are a fundamental concept. They are special vectors that, when a matrix is applied to them, do not change direction. Instead, they get scaled by a certain factor, which is the eigenvalue associated with that eigenvector. To put it simply, for a matrix \( A \), if \( \mathbf{v} \) is a non-zero vector that satisfies the equation \( A\mathbf{v} = \lambda\mathbf{v} \), where \( \lambda \) is a scalar (eigenvalue), then \( \mathbf{v} \) is an eigenvector of \( A \).
To find an eigenvector, we start by knowing the eigenvalue and then solving the equation \( (A - \lambda I)\mathbf{v} = 0 \), where \( I \) is the identity matrix of the same size as \( A \). This equation results in a system of linear equations that can be solved using methods like substitution or matrix row operations. An eigenvector, therefore, might not be unique but belongs to a whole line or plane of solutions, termed an eigenspace. It is crucial to understand that eigenvectors provide vital insights into the geometric transformations described by matrices.

Characteristic Equation

Every square matrix has a characteristic equation, which results from the determinant of \( A - \lambda I \), where \( A \) is the matrix and \( \lambda \) is a scalar. This equation is crucial in finding the eigenvalues of a matrix, as it is often written as \( \det(A - \lambda I) = 0 \).
This determinant results in a polynomial equation, often referred to as the characteristic polynomial. The roots of this characteristic polynomial give us the eigenvalues. For a matrix of size \( n \times n \), the characteristic polynomial is a polynomial of degree \( n \).
To solve the characteristic polynomial, you might look for rational roots, use graphical methods or apply the quadratic formula in simpler cases. These roots unlock the mystery of eigenvalues, allowing us to find the corresponding eigenvectors and better understand the transformation properties of a matrix.

Determinant

In linear algebra, the determinant of a square matrix is a scalar value that provides insight into the properties of the matrix. It can be helpful in understanding if a matrix is invertible, and plays a vital role in the calculation of eigenvalues. The determinant is the key component in the characteristic equation \( \det(A - \lambda I) = 0 \).
For a \(2 \times 2\) matrix, the determinant is calculated as \( ad - bc \). For larger matrices, the calculation involves more terms and may call for methods such as cofactor expansion.

• If the determinant of a matrix is zero, the matrix is not invertible.
• If the determinant is non-zero, the matrix is invertible.

This concept of the determinant forms the backbone of many operations in matrix algebra, particularly when it comes to solving systems of linear equations.

Matrix Algebra

Matrix algebra is a branch of mathematics that introduces matrices as tools to solve linear equations, perform transformations, and handle multi-dimensional data. A matrix is essentially a rectangular array of numbers organized in rows and columns. Two matrices can be added together, multiplied, and even find invertible matrices depending on their sizes.
Matrix multiplication is more complex than just multiplying corresponding elements. It involves taking the dot product of the rows of the first matrix with the columns of the second one. This operation is crucial in solving algebraic equations represented in matrix form and for transformations between geometric spaces.

• Matrices are indispensable in computer graphics, physics simulations, and any field requiring complex transformations and models.
• Basic operations include addition, subtraction, scalar multiplication, and finding the inverse and transpose of matrices.

Mastering matrix algebra equips you to tackle many real-world problems efficiently, making it an indispensable tool in scientific and engineering applications.