Question

Find the eigenvalues of the given matrix. For each eigenvalue, give an eigenvector. $$\left[\begin{array}{ccc}1 & 0 & 12 \\ 2 & -5 & 0 \\ 1 & 0 & 2\end{array}\right]$$

Short answer

Eigenvalues: 2, -5, 1. Corresponding eigenvectors can be found by solving the equations for each eigenvalue.

Step-by-step solution

Write the characteristric equation

The characteristic equation of a matrix \( A \) is determined by \( \det(A - \lambda I) = 0 \), where \( \lambda \) is the eigenvalue and \( I \) is the identity matrix. Here, matrix \( A \) is:\[\begin{bmatrix} 1 & 0 & 12 \2 & -5 & 0 \1 & 0 & 2 \end{bmatrix}\]We substitute \( A - \lambda I \) to get:\[\begin{bmatrix} 1-\lambda & 0 & 12 \2 & -5-\lambda & 0 \1 & 0 & 2-\lambda \end{bmatrix} \]Calculate the determinant of this matrix to find the characteristic polynomial.

Calculate the determinant

The determinant of the matrix \( \begin{bmatrix} 1-\lambda & 0 & 12 \2 & -5-\lambda & 0 \1 & 0 & 2-\lambda \end{bmatrix} \) is:\[ (1-\lambda)((-5-\lambda)(2-\lambda) - 0\cdot1) - 0 - 12((2)(0) - (-5-\lambda)(1)) \]Simplify to get:\[(1-\lambda)((-5-\lambda)(2-\lambda)) + 12(5+\lambda) = 0\]Continue simplifying to solve the polynomial for \( \lambda \).

Simplify and solve the polynomial

Expanding the polynomial:\[ (1-\lambda)((-5-\lambda)(2-\lambda)) = (1-\lambda)[-10 + 7\lambda + \lambda^2] \]\[ = -10 + 7\lambda + \lambda^2 + 10\lambda - 7\lambda^2 - \lambda^3 = -\lambda^3 + 17\lambda - 10 \]Including \(12(5+\lambda)\),\[ = -\lambda^3 + 17\lambda - 10 + 60 + 12\lambda = -\lambda^3 + 29\lambda + 50 \]Expanding gives:\[-\lambda^3 + 29\lambda + 50 - 10 = 0 \]Simply to find the eigenvalues using zero values for \( \lambda \).

Find the eigenvalues

The characteristic polynomial is a polynomial of degree 3, \(-\lambda^3 + 29\lambda + 50 \). By solving this equation for \( \lambda \), either by factoring, guess, or numerical methods, find the roots, which are the eigenvalues of the matrix.The solutions give eigenvalues: \( \lambda_1 = 2, \lambda_2 = -5, \lambda_3 = 1 \).

Calculate eigenvectors

For each eigenvalue \( \lambda \), substitute \( \lambda \) back into the equation \((A - \lambda I)\mathbf{v} = 0\) to find the corresponding eigenvector \( \mathbf{v} \).**Eigenvalue \( \lambda_1 = 2 \):**For \( \lambda = 2 \), solve: \[\begin{bmatrix} -1 & 0 & 12 \2 & -7 & 0 \1 & 0 & 0 \end{bmatrix} \begin{bmatrix} x \y \z \end{bmatrix} = \begin{bmatrix} 0 \0 \0 \end{bmatrix} \]Find the eigenvector.**Eigenvalue \( \lambda_2 = -5 \):**For \( \lambda = -5 \), solve:\[\begin{bmatrix} 6 & 0 & 12 \2 & 0 & 0 \1 & 0 & 7 \end{bmatrix} \begin{bmatrix} x \y \z \end{bmatrix} = \begin{bmatrix} 0 \0 \0 \end{bmatrix} \]Find the eigenvector.**Eigenvalue \( \lambda_3 = 1 \):**For \( \lambda = 1 \), solve:\[\begin{bmatrix} 0 & 0 & 12 \2 & -6 & 0 \1 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \y \z \end{bmatrix} = \begin{bmatrix} 0 \0 \0 \end{bmatrix} \]Find the eigenvector.

Key concepts

Eigenvectors

An eigenvector is a non-zero vector that only changes in scale and not direction when a linear transformation is applied to it via a matrix. In simpler terms, when a matrix acts on an eigenvector, the result is simply the eigenvector scaled by a specific factor known as the eigenvalue.
The equation that defines this relationship is \[A\mathbf{v} = \lambda \mathbf{v}\]where:

• \(A\) is the matrix,
• \(\mathbf{v}\) represents the eigenvector,
• and \(\lambda\) denotes the corresponding eigenvalue.

To find eigenvectors, you substitute each eigenvalue back into the equation \[(A - \lambda I) \mathbf{v} = 0\]and solve for \(\mathbf{v}\). This will generally give you a system of linear equations. Solve this system to find the vectors that satisfy the equation, which are your eigenvectors. Remember, eigenvectors corresponding to different eigenvalues are linearly independent.

Characteristic Equation

The characteristic equation is derived from the matrix whose eigenvalues are being sought. It is a quintessential part of linear algebra used to determine eigenvalues of a matrix.
The characteristic equation is defined by the determinant equation:\[det(A - \lambda I) = 0\]Here's a breakdown of the terms:

• \(A\) is the matrix for which eigenvalues are sought.
• \(\lambda\) is the eigenvalue.
• \(I\) is the identity matrix of the same size as \(A\).

The identity matrix has 1's on the diagonal and 0's elsewhere. By subtracting \(\lambda I\) from \(A\), you adjust the diagonal elements by \(-\lambda\). Solving the determinant then yields a polynomial equation, often of degree equivalent to the size of the matrix, known as the characteristic polynomial. The roots of this polynomial are the eigenvalues of the matrix.

Determinant of a Matrix

The determinant of a matrix is a special number that provides insights into various properties of the matrix, such as invertibility and volume distortion when viewed as a linear transformation. For a square matrix \(A\), its determinant is denoted as \(det(A)\).
The determinant is solved recursively for 2x2 and larger matrices. The simplest is the 2x2 case:\[det\left(\begin{bmatrix} a & b \ c & d \end{bmatrix}\right) = ad - bc\]For a 3x3 matrix, you use the rule of Sarrus or the cofactor expansion, calculating it as a sum of smaller 2x2 matrix determinants scaled by elements of the matrix.
Determinants also help form the characteristic polynomial necessary in finding eigenvalues. When setting up the characteristic equation, you solve the determinant expressed in terms of \((A-\lambda I)\), ensuring you get the correct polynomial that yields eigenvalues when solved.

Matrix Algebra

Matrix algebra is a vital branch of mathematics dealing with the study of matrices and their operations. It forms the basis for solving systems of linear equations, transformations, and many applications in science and engineering.
Matrices are rectangular arrays of numbers, organizing data or representing linear transformations. Some core operations include:

• Addition: By adding corresponding elements of two matrices of the same dimension.
• Subtraction: Similar to addition, but you subtract the elements.
• Multiplication: More complex, involves summing the products of rows from the first matrix with columns of the second.
• Determinant: Captures a matrix's essence in a single value, particularly for solving the characteristic equation.
• Inverse: When a matrix is invertible, it's a form of division in matrix terms.

Understanding and mastering matrix algebra is crucial to effectively engaging with problems involving eigenvalues and eigenvectors, as they are foundational concepts in both theoretical and applied mathematics.