Problem 49 - Calculus for Biology and Medicine

Question

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\), and graph the lines together with the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) and the vectors \(\mathrm{Av}_{1}\) and \(\mathrm{Av}_{2}\) $$A=\left[\begin{array}{rr}2 & 3 \\ 0 & -1\end{array}\right]$$

Step-by-Step Solution

Answer

Eigenvalues: \(\lambda_1 = 2\), \(\lambda_2 = -1\). Eigenvectors: \(\mathrm{v}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\), \(\mathrm{v}_2 = \begin{bmatrix} 1 \\ -1 \end{bmatrix}\).

Step 1: Find the Characteristic Equation of Matrix A

The characteristic equation is found by computing \(\det(A - \lambda I) = 0\). Here, \(I\) is the identity matrix, and \(\lambda\) is the eigenvalue we are solving for. Substitute \(A = \begin{bmatrix} 2 & 3 \ 0 & -1 \end{bmatrix}\) and \(I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\) to get \(A - \lambda I = \begin{bmatrix} 2-\lambda & 3 \ 0 & -1-\lambda \end{bmatrix}\). Calculate the determinant: \((2-\lambda)(-1-\lambda) - 0 \cdot 3 = \lambda^2 - \lambda - 2\). Set this equation to zero: \(\lambda^2 - \lambda - 2 = 0\).

Step 2: Solve for the Eigenvalues

Solve the quadratic equation \(\lambda^2 - \lambda - 2 = 0\). Factoring, we find \((\lambda - 2)(\lambda + 1) = 0\), so the eigenvalues are \(\lambda_1 = 2\) and \(\lambda_2 = -1\).

Step 3: Find the Eigenvectors

For \(\lambda_1 = 2\), solve \((A - 2I)\mathrm{v}_1 = 0\). Substitute to get \(\begin{bmatrix} 0 & 3 \ 0 & -3 \end{bmatrix}\begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix}\). This simplifies to \(3y = 0\), giving \(y = 0\) and \(x\) is any value. The eigenvector is \(\mathrm{v}_1 = \begin{bmatrix} 1 \ 0 \end{bmatrix}\). For \(\lambda_2 = -1\), solve \((A + I)\mathrm{v}_2 = 0\). Substitute to get \(\begin{bmatrix} 3 & 3 \ 0 & 0 \end{bmatrix}\begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix}\). This gives \(3x + 3y = 0\) or \(x = -y\). The eigenvector can be \(\mathrm{v}_2 = \begin{bmatrix} 1 \ -1 \end{bmatrix}\).

Step 4: Determine Equations of Lines through Origin

The lines through the origin can be described by the direction of the eigenvectors. For \(\mathrm{v}_1 = \begin{bmatrix} 1 \ 0 \end{bmatrix}\), the line equation is \(y = 0\). For \(\mathrm{v}_2 = \begin{bmatrix} 1 \ -1 \end{bmatrix}\), the line equation is \(y = -x\).

Step 5: Graph the Lines and Vectors

Plot the line \(y = 0\) which runs along the \(x\)-axis and the line \(y = -x\), which passes through the origin at a \(-45\) degree angle from the \(x\)-axis. Plot \(\mathrm{v}_1 = \begin{bmatrix} 1 \ 0 \end{bmatrix}\) as a vector along the \(x\)-axis, and \(\mathrm{v}_2 = \begin{bmatrix} 1 \ -1 \end{bmatrix}\) in the direction of \(y = -x\). Calculate \(A\mathrm{v}_1 = \begin{bmatrix} 2 & 3 \ 0 & -1 \end{bmatrix}\begin{bmatrix} 1 \ 0 \end{bmatrix} = \begin{bmatrix} 2 \ 0 \end{bmatrix}\) and \(A\mathrm{v}_2 = \begin{bmatrix} 2 & 3 \ 0 & -1 \end{bmatrix}\begin{bmatrix} 1 \ -1 \end{bmatrix} = \begin{bmatrix} -1 \ 1 \end{bmatrix}\), and plot these vectors. The \(A\mathrm{v}_1\) overlies \(\mathrm{v}_1\) enlarging the \(x\)-component, and \(A\mathrm{v}_2\) is a reflection of \(\mathrm{v}_2\.\)

Key Concepts

Characteristic Equation

The characteristic equation is fundamental when working with eigenvalues and eigenvectors. It's derived from subtracting a scalar multiple of the identity matrix \(I\) from a given matrix \(A\), and setting the determinant of this matrix equal to zero. This scalar is represented by the variable \(\lambda\), which stands for the eigenvalues.

The equation \(\det(A - \lambda I) = 0\) is what you're trying to solve. For the matrix \(A = \begin{bmatrix} 2 & 3 \ 0 & -1 \end{bmatrix}\), subtracting \(\lambda I\) gives us \(\begin{bmatrix} 2-\lambda & 3 \ 0 & -1-\lambda \end{bmatrix}\). Computing the determinant results in a quadratic equation \(\lambda^2 - \lambda - 2 = 0\).

• Determinant as the setting for finding eigenvalues.
• Creating a polynomial equation that reveals the eigenvalues.

Determinant

The determinant plays a key role in assessing whether a matrix has eigenvalues, which are essentially the roots of the characteristic equation.

For a 2x2 matrix like \(A\), the determinant is calculated by the formula \(ad - bc\), where \(a, b, c,\) and \(d\) are elements of the matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\). For our matrix \(A = \begin{bmatrix} 2 & 3 \ 0 & -1 \end{bmatrix}\), the determinant of \(A - \lambda I\) results in \(\lambda^2 - \lambda - 2\).

• The determinant must be zero for eigenvalues to exist.
• Leads to a solvable polynomial for \(\lambda\) when substituted correctly.

Matrix Operations

Matrix operations involve a variety of mathematical processes that transform matrices and help solve for eigenvalues and eigenvectors.

One primary operation here is the subtraction of matrices, necessary for forming \(A - \lambda I\). In this case, subtracting \(\lambda\) times the identity matrix from \(A\). Additionally, multiplying matrices also comes into play when finding eigenvectors. After finding each eigenvalue, we plug it back into \(A - \lambda I\) to obtain a system of linear equations.

• Subtraction is used to adjust the original matrix by \(\lambda I\).
• Matrix multiplication forms part of verifying relationships between eigenvectors.

Linear Transformations

Linear transformations are functions that map vectors to other vectors in a linear manner. Each linear transformation can be represented by a matrix, and this matrix defines how input vectors are transformed.

Through the process of analyzing eigenvectors and eigenvalues, we basically assess how certain directions (or eigenvectors) are stretched or shrunk by the transformation matrix. In our example with matrix \(A\), eigenvectors \(\mathrm{v}_1 = \begin{bmatrix} 1 \ 0 \end{bmatrix}\) and \(\mathrm{v}_2 = \begin{bmatrix} 1 \ -1 \end{bmatrix}\) indicated directions that are either unchanged in direction or scaled by factors given by eigenvalues.

• Identifies directions that remain invariant under transformation.
• Shows how these directions are scaled, compressed, or expanded.