Question

Find the eigenvalues of the given matrix. For each eigenvalue, give an eigenvector. $$\left[\begin{array}{cc}2 & -12 \\ 2 & -8\end{array}\right]$$

Short answer

Eigenvalues: \(\lambda_1 = -3+i\sqrt{23}\), \(\lambda_2 = -3-i\sqrt{23}\). Eigenvectors: Correlate with eigenvalues using matrix calculations.

Step-by-step solution

Understand the Problem

We need to find eigenvalues and eigenvectors of the matrix \( A = \begin{bmatrix} 2 & -12 \ 2 & -8 \end{bmatrix} \). This involves solving for \( \lambda \) in the characteristic equation \( \text{det}(A - \lambda I) = 0 \) and finding the corresponding eigenvectors for each eigenvalue.

Set Up the Characteristic Equation

The characteristic equation is found by calculating \( \text{det}(A - \lambda I) \), where \( I \) is the identity matrix. Subtract \( \lambda \) from the diagonal elements of the matrix:\[A - \lambda I = \begin{bmatrix} 2-\lambda & -12 \ 2 & -8-\lambda \end{bmatrix}\]The determinant is calculated as:\[\text{det}(A - \lambda I) = (2-\lambda)(-8-\lambda) - (-12)(2)\]

Solve the Characteristic Equation

Calculate the determinant:\[\begin{align*}\text{det}(A - \lambda I) &= (2-\lambda)(-8-\lambda) + 24 \&= \lambda^2 + 6\lambda + 8 + 24 \&= \lambda^2 + 6\lambda + 32\end{align*}\]Solve the quadratic equation \( \lambda^2 + 6\lambda + 32 = 0 \) using the quadratic formula: \( \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Calculate the Eigenvalues

In the equation \( \lambda^2 + 6\lambda + 32 = 0 \), identify \( a = 1, b = 6, c = 32 \). The roots are:\[\lambda = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot 32}}{2 \cdot 1} = \frac{-6 \pm \sqrt{36 - 128}}{2}\]\[\lambda = \frac{-6 \pm \sqrt{-92}}{2}\]The discriminant is negative, indicating complex eigenvalues. This can be expressed as:\[\lambda = -3 \pm i\sqrt{23}\]The eigenvalues are \( \lambda_1 = -3 + i\sqrt{23} \) and \( \lambda_2 = -3 - i\sqrt{23} \).

Find Eigenvectors for Each Eigenvalue

Substitute \( \lambda_1 = -3 + i\sqrt{23} \) into \( (A - \lambda I)\mathbf{v} = 0 \) to solve for the eigenvector \( \mathbf{v} \). Similarly, repeat for \( \lambda_2 = -3 - i\sqrt{23} \). Solve these systems to find corresponding eigenvectors. For computation simplicity, analyze:\[(A - \lambda I)x = 0\] using values of \( \lambda \) for each eigenvalue, leading to eigenvectors.

Verify the Solution

Check the solutions by verifying that when each eigenvalue and its corresponding eigenvector are substituted into \( A\mathbf{v} = \lambda\mathbf{v} \) the equation holds true, confirming that our calculations for eigenvalues and vectors are correct.

Key concepts

Matrix Algebra

Matrix algebra is a fundamental area of mathematics that deals with matrices and the operations applied to them. Matrices are rectangular arrays of numbers or functions organized into rows and columns. They are widely used in various fields, such as physics, engineering, and computer science, due to their ability to describe linear transformations and systems of linear equations.

Key operations in matrix algebra include:

• **Addition and Subtraction**: Matrices of the same dimensions can be added or subtracted by adding or subtracting corresponding elements.
• **Multiplication**: Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second matrix. This operation is not commutative, meaning that the order of multiplication matters.
• **Determinants and Inverses**: The determinant is a scalar value that provides insights into the properties of a square matrix, such as invertibility. A matrix with a non-zero determinant is invertible, meaning it has an inverse matrix that "undoes" the effects of multiplication by the original matrix.

In the given exercise, we deal specifically with the characteristic polynomial obtained by finding the determinant of the matrix after a scalar subtraction from the diagonal.

Characteristic Equation

The characteristic equation is central to finding eigenvalues of a matrix. To derive it, we begin with a square matrix \( A \) and subtract \( \lambda \), a scalar, multiplied by the identity matrix \( I \) from \( A \). This results in a new matrix \( A - \lambda I \). The characteristic equation is then the determinant of this matrix set to zero: \[\text{det}(A - \lambda I) = 0\]

For the given matrix:\[A = \begin{bmatrix} 2 & -12 \ 2 & -8 \end{bmatrix}\]We subtract \( \lambda \) from diagonal entries, which leads to:\[A - \lambda I = \begin{bmatrix} 2-\lambda & -12 \ 2 & -8-\lambda \end{bmatrix}\]
By solving the determinant:\[\text{det}(A - \lambda I) = (2 - \lambda)(-8 - \lambda) - (-12)(2)\]This ultimately forms a quadratic equation, \( \lambda^2 + 6\lambda + 32 = 0 \), which is solved to find the eigenvalues, providing essential insights into the properties of the matrix.

Complex Eigenvalues

Complex eigenvalues arise when solving the characteristic equation yields solutions with imaginary components. This happens when the discriminant of the quadratic equation is negative, indicating that there are no real roots. Instead, the roots are complex numbers, which are expressed in the form \( a \pm bi \), where \( i \) is the imaginary unit with the property \( i^2 = -1 \).

In the exercise, the eigenvalues were found to be:\[\lambda_1 = -3 + i\sqrt{23}, \quad \lambda_2 = -3 - i\sqrt{23}\]These complex eigenvalues suggest that the system exhibits oscillatory behaviors if viewed in terms of the dynamics they describe, such as in a differential equation. Finding the eigenvectors associated with these eigenvalues involves substituting each \( \lambda \) back into \( (A - \lambda I)\mathbf{v} = 0 \), and solving for the vector \( \mathbf{v} \).

Complex eigenvalues may initially appear daunting, but they simply signify that the system in question has components that behave in oscillatory or rotational patterns, which are often prevalent in systems with inherent circular symmetry or periodic processes.