Question

A matrix \(A\) and one of its eigenvalues are given. Find an eigenvector of A for the given eigenvalue. $$ \begin{aligned} A &=\left[\begin{array}{ccc} -16 & -28 & -19 \\ 42 & 69 & 46 \\ -42 & -72 & -49 \end{array}\right] \\ \lambda &=5 \end{aligned} $$

Short answer

The eigenvector for \( \lambda = 5 \) is \(\begin{bmatrix} 3 \\ -7 \\ 7 \end{bmatrix}\).

Step-by-step solution

Understand the Eigenvalue Equation

To find an eigenvector corresponding to the eigenvalue \( \lambda = 5 \), we need to solve the equation \( A\mathbf{v} = \lambda \mathbf{v} \) for the vector \( \mathbf{v} \). This simplifies to \( (A - \lambda I)\mathbf{v} = \mathbf{0} \), where \( I \) is the identity matrix.

Formulate the Equation

Substitute \( \lambda = 5 \) and the identity matrix into the equation. We calculate \( A - 5I \), where \( I \) is the \( 3 \times 3 \) identity matrix.

Calculate \( A - 5I \)

Subtract 5 from the diagonal elements of matrix \( A \): \[A - 5I = \begin{bmatrix}-21 & -28 & -19 \42 & 64 & 46 \-42 & -72 & -54\end{bmatrix}\]

Solve the Homogeneous Equation

Next, solve the equation \((A - 5I)\mathbf{v} = \mathbf{0} \). This can be done by finding the null space of the matrix \( A - 5I \), which involves row reducing the matrix.

Row-Reduce the Matrix

Perform row operations to row-reduce the matrix \( A - 5I \):1. Multiply the first row by 2 and add to the second row:\[\begin{bmatrix}-21 & -28 & -19 \0 & 8 & 8 \-42 & -72 & -54\end{bmatrix}\]2. Multiply the first row by -2 and add to the third row:\[\begin{bmatrix}-21 & -28 & -19 \0 & 8 & 8 \0 & -16 & -16\end{bmatrix}\]3. Multiply the second row by 2 and add to the third row:\[\begin{bmatrix}-21 & -28 & -19 \0 & 8 & 8 \0 & 0 & 0\end{bmatrix}\]

Interpret Row-Reduced Form

The row-reduced form shows us the following system:1. \(-21x_1 - 28x_2 - 19x_3 = 0\)2. \(8x_2 + 8x_3 = 0\)We can express \(x_2 = -x_3\).

Find Eigenvector

We substitute \(x_2 = -x_3\) into the first equation: \(-21x_1 - 28(-x_3) - 19x_3 = 0 \), simplifying to \(-21x_1 + 9x_3 = 0\).Thus, \(x_1 = \frac{3x_3}{7}\).Choosing \(x_3 = 7\), we get \(x_1 = 3\) and \(x_2 = -7\).Thus, an eigenvector corresponding to the eigenvalue 5 is \(\begin{bmatrix} 3 \ -7 \ 7 \end{bmatrix}\).

Key concepts

Matrix Algebra

Matrix algebra serves as the foundation for various calculations and concepts within linear algebra, including operations such as addition, subtraction, and multiplication of matrices. These operations help solve complex systems of linear equations and other mathematical models.
Matrices are rectangular arrays of numbers arranged in rows and columns and are often used to represent linear mappings or transformations in space. Understanding the behavior of matrices is crucial for solving problems related to eigenvalues and eigenvectors.

• Addition and Subtraction: Matrices of the same dimensions can be added or subtracted by corresponding elements.
• Multiplication: The product of two matrices is another matrix, contributed by the sum of products of the entries.
• Identity Matrix: The identity matrix, denoted as \( I \), is a special type of matrix where all the diagonal elements are 1, and others are 0, serving a role similar to the number 1 in real numbers multiplication.

Mastery in matrix algebra allows us to tackle eigenvalue and eigenvector problems efficiently, crucial for many applications in science and engineering.

Eigenvalue Equation

The eigenvalue equation is fundamental in understanding the attributes of a matrix concerning its transformations and exists within the formula: \( A\mathbf{v} = \lambda \mathbf{v} \). This equation seeks a scalar value \( \lambda \) known as the eigenvalue, which, when multiplied by its associated vector \( \mathbf{v} \) (known as the eigenvector), leaves the direction of \( \mathbf{v} \) unchanged.
By rearranging the terms, we derive another form: \((A - \lambda I) \mathbf{v} = \mathbf{0} \). This expression highlights how the matrix \( A \) interacts with the identity matrix \( I \) to yield zero through certain vectors.

• The change induced by \( A \) replicates the effect of multiplying by \( \lambda \), the scalar.
• \( \lambda \) can often inform us about the properties of \( A \), such as stability and oscillation in systems.
• This concept is pivotal for problems in quantum mechanics, vibration analysis, and principal component analysis.

Solving the eigenvalue equation leads to discovering inherent properties of the system represented by the matrix, providing powerful insights and solutions.

Row Reduction

Row reduction, also known as Gaussian elimination, is a method used to simplify matrices to make the solution of systems of equations straightforward. It is particularly useful in finding eigenvectors once we have calculated \( (A - \lambda I) \). By reducing the system to a simpler or even diagonal form, we can easily observe the relationships between variables.
The process involves systematic operations:

• Swapping Rows: To bring a non-zero row into a pivotal position.
• Multiplying rows: By a non-zero scalar to adjust pivotal coefficients.
• Adding/Subtracting rows: To eliminate a variable from other rows.

The goal is to prepare the matrix in a form known as Row-Echelon form or further into Reduced Row-Echelon form, where the solution becomes transparent.
Row reduction applied to \( A - \lambda I \) eliminates excess variables, reducing the complexity of interpreting eigenvectors, ultimately leading to clear solutions and interpretations.

Homogeneous Equations

Homogeneous equations play a critical role when dealing with eigenvalue problems, as seen in the last step of solving for eigenvectors. These equations are characterized by an equal sign leading to zero on the right-hand side, for example, \( (A - \lambda I)\mathbf{v} = \mathbf{0} \). They are fundamental in finding the null space of a matrix.
When an equation is homogeneous, it often indicates that the resultant vector from the matrix transformation does not change direction, consistent with eigenvector behavior. Solving such equations involves:

• Recognizing the trivial solution, \( \mathbf{v} = \mathbf{0} \), is always possible but often not useful.
• Finding non-trivial solutions by determining free variables or dependencies among variables.

The method is crucial for discovering eigenvectors, pivotal in understanding directional properties of transformations by matrix \( A \), offering valuable insights into its comprehensive behavioral features and solutions tailored to applied mathematical scenarios.