Question

Find the eigenvalues of the given matrix. For each eigenvalue, give an eigenvector. $$\left[\begin{array}{cc}5 & 9 \\ -1 & -5\end{array}\right]$$

Short answer

Eigenvalues are \(\sqrt{11}\) and \(-\sqrt{11}\); corresponding eigenvectors are \([9, \sqrt{11} - 5]\) and \([9, -5-\sqrt{11}]\).

Step-by-step solution

Understand the Problem

We need to find the eigenvalues and the corresponding eigenvectors for the given 2x2 matrix. Eigenvalues are solutions to the characteristic equation, and eigenvectors are non-zero vectors associated with these eigenvalues.

Set Up the Characteristic Equation

The characteristic equation is derived from the determinant of the matrix \(A - \lambda I\), where \(A\) is the given matrix, \(\lambda\) is the eigenvalue, and \(I\) is the identity matrix. For our matrix, this is: \[\text{det}\left(\begin{bmatrix} 5 - \lambda & 9 \ -1 & -5 - \lambda \end{bmatrix}\right) = 0\].

Calculate the Determinant

Calculate the determinant of matrix \(\begin{bmatrix} 5-\lambda & 9 \ -1 & -5-\lambda \end{bmatrix}\). The determinant is: \[(5-\lambda)(-5-\lambda) - (9)(-1) = \lambda^2 - 0\lambda - 20 + 9.\]The characteristic polynomial simplifies to \(\lambda^2 - 11\).

Solve for the Eigenvalues

Solve the characteristic equation \(\lambda^2 - 11 = 0\). The solutions, or eigenvalues, are found by solving \(\lambda^2 = 11\), yielding \(\lambda = \sqrt{11}\) and \(\lambda = -\sqrt{11}\).

Find Eigenvector for \(\lambda = \sqrt{11}\)

Substitute \(\lambda = \sqrt{11}\) into \((A - \lambda I)\mathbf{x} = 0\). The matrix becomes \(\begin{bmatrix} 5 - \sqrt{11} & 9 \ -1 & -5 - \sqrt{11} \end{bmatrix}\). The eigenvector \(\mathbf{x}\) corresponds typically to the ratios of entries solving the system. By simplifying, choose \(x_1 = 9\), \(x_2 = \sqrt{11}-5\).

Find Eigenvector for \(\lambda = -\sqrt{11}\)

Substitute \(\lambda = -\sqrt{11}\) into \((A - \lambda I)\mathbf{x} = 0\). The matrix becomes:\(\begin{bmatrix} 5 + \sqrt{11} & 9 \ -1 & -5 + \sqrt{11} \end{bmatrix}\). The eigenvector \(\mathbf{x}\) by consistent entries is another set of ratios.Choose \(x_1 = 9\), \(x_2 = -5-\sqrt{11}\).

Verify the Eigenvectors

Ensure that the computed eigenvectors satisfy the equation \((A - \lambda I)\mathbf{x} = 0\). This step involves inserting the vectors back into the original matrix equation and performing matrix-vector multiplication to check if it results in a zero vector.

Key concepts

Characteristic Equation

The characteristic equation plays a vital role in finding eigenvalues of a matrix. It is derived from the expression \(\text{det}(A - \lambda I) = 0\), where \(A\) is the original matrix, \(\lambda\) represents potential eigenvalues, and \(I\) is the identity matrix of the same size as \(A\). By setting this determinant equal to zero, we form a polynomial equation, typically in terms of \(\lambda\), which we solve to find the eigenvalues.
For a 2x2 matrix such as \(\begin{bmatrix}5 & 9 \ -1 & -5 \end{bmatrix}\), the characteristic equation becomes \(\text{det}(\begin{bmatrix}5-\lambda & 9 \ -1 & -5-\lambda\end{bmatrix}) = 0\).
This means you need to perform the determinants' arithmetic to end up with a simple polynomial, which, in this example, results in \(\lambda^2 - 11 = 0\). By solving this polynomial, you find the values of \(\lambda\) that satisfy the equation, in this case, \(\lambda = \pm\sqrt{11}\).

Determinant of a Matrix

The determinant of a matrix is a special number that can be computed from its elements. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated as \(ad - bc\).
The determinant gives us important information about the matrix, such as whether it has an inverse (non-zero determinant means the matrix is invertible) and plays a crucial role in defining the characteristic equation.
In our discussed problem, we find the determinant of \(A - \lambda I\) which is \((5-\lambda)(-5-\lambda) - (-1)(9)\), simplifying to \(\lambda^2 - 11\).
This polynomial helps pinpoint the eigenvalues, with each value representing a necessary condition for an associated matrix transformation.

2x2 Matrices

2x2 matrices are the simplest form of square matrices and serve as a fundamental starting point for understanding larger matrices. These matrices are important because they fit within a two-dimensional space and often represent linear transformations such as rotation, scaling, or shearing.
In situations involving eigenvalues and eigenvectors, a 2x2 matrix provides a concise way to explore these concepts. The math involved is typically more straightforward compared to higher-dimensional matrices, making 2x2 matrices ideal for gaining a preliminary understanding of matrix algebra concepts like determinants and characteristic equations.
The matrix \(\begin{bmatrix} 5 & 9 \ -1 & -5 \end{bmatrix}\) is a perfect example to practice on, as solving for eigenvalues and their respective eigenvectors can clearly illustrate the process without overwhelming complexity.
Understanding 2x2 matrices thoroughly paves the way for more advanced topics in linear algebra.

Matrix Algebra

Matrix algebra encompasses a wide array of operations that can be performed on matrices, such as addition, multiplication, and taking determinants.
This form of algebra is crucial in many fields, including computer science, economics, and engineering, because it provides tools to work with linear equations and transformations more effectively.
In the context of our problem, matrix algebra enables the transformation of the original matrix through the expression \(A - \lambda I\), simplifying calculations by reducing the problem to smaller, manageable components.
This transformation is key when deriving the characteristic equation, allowing us to identify eigenvalues and eigenvectors. Through operations like determining \(\det(A - \lambda I)\), matrix algebra serves as the backbone in solving numerous mathematical problems and real-world applications.