Problem 56 - Calculus for Biology and Medicine

Question

In Problems , find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), and graph the lines together with the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) and the vectors \(A \mathbf{v}_{1}\) and \(A \mathbf{v}_{2}\). $$ A=\left[\begin{array}{rr} -3 & -0.5 \\ 7 & 1.5 \end{array}\right] $$

Step-by-Step Solution

Answer

Eigenvalues are complex; compute eigenvectors from \( (A - \lambda I)\mathbf{v} = 0 \). Plot lines and vectors accordingly.

Step 1: Find the Characteristic Equation

To find the eigenvalues, we need to determine the roots of the characteristic equation \( \det(A - \lambda I) = 0 \). For the matrix \( A \), \( I \) is the identity matrix. Hence, the characteristic matrix is:\[ A - \lambda I = \begin{bmatrix} -3 - \lambda & -0.5 \ 7 & 1.5 - \lambda \end{bmatrix} \]

Step 2: Solve the Determinant Equation

Next, compute the determinant of \( A - \lambda I \):\[ \det(A - \lambda I) = (-3 - \lambda)(1.5 - \lambda) - (7)(-0.5) \] Simplifying this, \[ \det(A - \lambda I) = (\lambda + 3)(\lambda - 1.5) + 3.5 = \lambda^2 + 1.5\lambda + 4 \] Set the determinant equal to zero to get the characteristic equation: \[ \lambda^2 + 1.5\lambda + 4 = 0 \]

Step 3: Solve the Quadratic Equation

Solve the quadratic equation \( \lambda^2 + 1.5\lambda + 4 = 0 \) using the quadratic formula \( \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 1.5, \) and \( c = 4 \). The discriminant \( b^2 - 4ac = 1.5^2 - 4 \cdot 1 \cdot 4 = -13.75 \). Since the discriminant is negative, the eigenvalues are complex: \[ \lambda_1, \lambda_2 = \frac{-1.5 \pm i\sqrt{13.75}}{2} \]

Step 4: Find the Eigenvectors

For each eigenvalue, solve \( (A - \lambda I)\mathbf{v} = 0 \). Using \( \lambda_1 = \frac{-1.5 + i\sqrt{13.75}}{2} \) and similar for \( \lambda_2 \). As the calculations involve complex numbers, assume a vector form and solve for components resulting in eigenvectors (Example: \( \mathbf{v}_1 = \begin{bmatrix} 1 \ -1.25 \end{bmatrix}\)).

Step 5: Describe the Lines Through the Origin

The direction of the lines through the origin is given by the eigenvectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \). The equations are parameterized as \( \mathbf{r} = t\mathbf{v}_i \) for each eigenvector, where \( t \) is a parameter.

Step 6: Graph Eigenvectors, Transformed Vectors, and Lines

Plot the eigenvectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \) from the origin. Compute \( A\mathbf{v}_1 \) and \( A\mathbf{v}_2 \) to find the transformed vectors. Graph these vectors alongside the eigenvectors and their corresponding lines through the origin using vector plotting techniques.

Key Concepts

Characteristic Equation

When dealing with matrices and their eigenvalues, a central piece of the puzzle is the characteristic equation. This equation emerges from the expression \( \det(A - \lambda I) = 0 \), where \( A \) is your matrix, \( \lambda \) represents the eigenvalues you're solving for, and \( I \) is the identity matrix. Essentially, you modify the matrix \( A \) by subtracting \( \lambda \) (the variable for eigenvalues) times the identity matrix, \( I \), creating a new matrix \( A - \lambda I \).

The determinant of this new matrix is what forms the characteristic equation. Solving this equation involves finding values of \( \lambda \) that make this determinant zero. These values are your eigenvalues, which are crucial in understanding and processing the properties of the transformation represented by matrix \( A \).

• Eigenvalues reveal essential characteristics of matrix transformations.
• The characteristic equation is central to finding these eigenvalues.
• It connects matrix algebra with polynomial equations.

Understanding this equation is crucial as it sets the stage for determining the stability and nature of dynamic systems.

Determinant

The determinant is a numerical value derived from a square matrix. It provides significant insight into the matrix's properties, such as invertibility and scaling factor. In the context of finding eigenvalues, the determinant plays a pivotal role in constructing the characteristic equation.

For a 2x2 matrix, like our matrix \( A \), the determinant is calculated by multiplying the elements on the main diagonal and then subtracting the product of the off-diagonal elements. So, for the matrix \( A - \lambda I \), the determinant is computed as:
\[det(A - \lambda I) = (-3 - \lambda)(1.5 - \lambda) - (7)(-0.5)\]
The determinant must equal zero to find the eigenvalues, thus forming part of the characteristic equation, \( \lambda^2 + 1.5\lambda + 4 = 0 \) in this case.

• Solving this indicates points where a linear transformation does not change areas in the plane (areas collapse to zero).
• The determinant helps gauge the overall effect of the linear transformation described by matrices.

By understanding determinants, you build a bridge to more complex topics like eigenvectors and eigenvalues.

Quadratic Formula

A crucial method for finding the roots of quadratic equations is the quadratic formula. This formula states:
\[\lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here, \( a \), \( b \), and \( c \) are coefficients from a quadratic equation of the form \( a\lambda^2 + b\lambda + c = 0 \). In our characteristic equation \( \lambda^2 + 1.5\lambda + 4 = 0 \), these coefficients are \( a = 1 \), \( b = 1.5 \), and \( c = 4 \).

The expression under the square root, \( b^2 - 4ac \), is called the discriminant. It indicates the nature of the roots:

• If positive, there are two distinct real roots.
• If zero, there's exactly one real root.
• If negative, as in our case, the roots are complex numbers.

Understanding the quadratic formula and the discriminant is pivotal because it allows you to predict the behavior of solutions, especially when these solutions venture into the realm of complex numbers.

Complex Numbers

In mathematics, complex numbers extend the concept of one-dimensional numbers (real numbers) into two dimensions. It introduces an imaginary unit \( i \), where \( i^2 = -1 \). Complex numbers take the form \( a + bi \), where \( a \) and \( b \) are real numbers.

In our context, the eigenvalues are complex numbers because the discriminant in the quadratic formula was negative, resulting in roots \( \lambda = \frac{-1.5 \pm i\sqrt{13.75}}{2} \).

• Complex numbers include a real part and an imaginary part.
• They provide solutions for cases where no real number solution exists.
• Within matrices, they help describe rotations and oscillations that real numbers cannot.

Although they might seem perplexing initially, complex numbers are essential in many fields such as engineering, physics, and applied mathematics, providing a more comprehensive view of problems that real numbers alone cannot express.