Problem 11 - Advanced Engineering Mathematics

Question

In Problems 7-22, find the eigenvalues and eigenvectors of the given matrix. Using Theorem 8.8.2 or (6), state whether the matrix is singular or nonsingular. $$ \left(\begin{array}{ll} -1 & 2 \\ -5 & 1 \end{array}\right) $$

Step-by-Step Solution

Answer

The eigenvalues are \( \frac{-1 \pm \sqrt{37}}{2} \) and the matrix is nonsingular.

Step 1: Define Eigenvalue Equation

The eigenvalue equation for a matrix \( A \) is \( A\mathbf{v} = \lambda\mathbf{v} \). This can be rewritten as \( A - \lambda I = 0 \), where \( I \) is the identity matrix. Thus, to find the eigenvalues, we compute the determinant of \( A - \lambda I \).

Step 2: Subtract \( \lambda \) from Diagonal Elements

The matrix \( A - \lambda I \) becomes: \[\begin{pmatrix} -1 - \lambda & 2 \ -5 & 1 - \lambda \end{pmatrix}\]

Step 3: Calculate Determinant

Compute the determinant of \( A - \lambda I \): \[\det\begin{pmatrix} -1 - \lambda & 2 \ -5 & 1 - \lambda \end{pmatrix} = (-1 - \lambda)(1 - \lambda) - (2)(-5)\]This simplifies to \[\lambda^2 + \lambda - 9 = 0\]

Step 4: Solve Characteristic Equation

Solve the quadratic equation \( \lambda^2 + \lambda - 9 = 0 \) to find the eigenvalues. Use the quadratic formula \( \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 1 \), and \( c = -9 \).

Step 5: Compute Eigenvalues

Plug the coefficients into the quadratic formula: \[\lambda = \frac{-1 \pm \sqrt{1 + 36}}{2}\]\[\lambda = \frac{-1 \pm \sqrt{37}}{2}\]The eigenvalues are: \( \lambda_1 = \frac{-1 + \sqrt{37}}{2} \) and \( \lambda_2 = \frac{-1 - \sqrt{37}}{2} \).

Step 6: Determine Non-singularity

A matrix is nonsingular if it has no zero eigenvalues. Since both eigenvalues, \( \frac{-1 + \sqrt{37}}{2} \) and \( \frac{-1 - \sqrt{37}}{2} \), are nonzero, the matrix is nonsingular.

Key Concepts

Matrix Operations

Understanding matrix operations is crucial when dealing with eigenvalues and eigenvectors, as they involve performing specific algebraic actions on matrices. Basic matrix operations include addition and subtraction, which combine matrices of the same dimension by adding or subtracting their corresponding elements.
Multiplication is more complex and involves taking the dot product of rows and columns, which requires careful alignment of matrix dimensions and results in a new matrix. Each element in the product results from summing the products of elements from respective rows and columns. Other operations like scalar multiplication apply a single number across every element of the matrix.
Two special matrix operations vital for eigenvalues and eigenvectors are transposition and finding the inverse. The transpose of a matrix switches its rows with its columns, which can be especially useful in various linear algebra applications. Finding the inverse is more complex; it only exists for nonsingular matrices and performs the role of an algebraic inverse, but for matrices.

• Transposition - flipping rows and columns.
• Addition/Subtraction - element-wise operations for matrices of same size.
• Multiplication - aligning rows of first matrix with columns of second.
• Inverse - exists only for nonsingular matrices.

Eigenvalue Equation

At the heart of finding eigenvalues is the eigenvalue equation, a fundamental component of linear algebra. It describes a special relationship between a square matrix and a scalar, known as the eigenvalue. The equation is given as:
\[A\mathbf{v} = \lambda\mathbf{v}\] where:

• \( A \) is a matrix,
• \( \mathbf{v} \) is an eigenvector corresponding to the eigenvalue,
• \( \lambda \) represents the eigenvalue.

This equation illustrates how matrix multiplication can stretch or compress eigenvectors without changing their direction. To find the eigenvalues, you must rewrite this equation in a slightly different form:
\[(A - \lambda I)\mathbf{v} = 0\]Here, \( I \) is the identity matrix, and \( (A - \lambda I) \) is a singular matrix. The objective is to find values of \( \lambda \) that make this equation true, meaning there is a nontrivial solution for \( \mathbf{v} \). This involves solving the characteristic equation derived from the determinant of \( A - \lambda I \).

Determinant Calculation

Calculating the determinant is a critical step in finding eigenvalues. The determinant helps in checking the invertibility of a matrix and gives insight into solving the eigenvalue equation.
The determinant of a 2x2 matrix such as:
\[\begin{pmatrix} a & b \ c & d \end{pmatrix}\]is calculated as:\[ad - bc\]This value determines if a matrix is singular (result is zero) or nonsingular (result is non-zero).
In the context of eigenvalues, you calculate the determinant of \( A - \lambda I \) to set up the characteristic polynomial. For the example matrix:
\[A - \lambda I =\begin{pmatrix} -1 - \lambda & 2 \ -5 & 1 - \lambda \end{pmatrix}\],you compute its determinant as \[(-1-\lambda)(1-\lambda) - (2)(-5)\].
The roots of the resulting polynomial give us the eigenvalues.

Nonsingular Matrix

A nonsingular matrix is one that has an inverse and is typically important in various linear algebra applications. This characteristic is directly related to its eigenvalues; specifically, a matrix is nonsingular if none of its eigenvalues are zero. This is because having a zero eigenvalue indicates the matrix is not invertible, as its determinant would also be zero.
To identify whether a matrix like\[\begin{pmatrix} -1 & 2 \ -5 & 1 \end{pmatrix}\]is nonsingular, we look at its eigenvalues. Through the solution process, we find that the eigenvalues are \(\lambda_1 = \frac{-1 + \sqrt{37}}{2}\)and \(\lambda_2 = \frac{-1 - \sqrt{37}}{2}\).
Both of these values are non-zero, confirming that the matrix is indeed nonsingular. This property implies robustness in mathematical systems since nonsingular matrices can be safely inverted or used in solving systems of equations. Nonsingularity guarantees that the matrix functions well in assignments such as transforming vectors without collapsing them into lower-dimensional forms.