Problem 62 - Calculus for Biology and Medicine

Question

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for $$ A=\left[\begin{array}{ll} a & c \\ 0 & b \end{array}\right] $$

Step-by-Step Solution

Answer

The eigenvalues are \( \lambda_1 = a \) and \( \lambda_2 = b \).

Step 1: Understand the Eigenvalue Equation

To find the eigenvalues of a matrix, we solve the characteristic equation \( \text{det}(A - \lambda I) = 0 \). Here, \( I \) is the identity matrix of the same size as \( A \).

Step 2: Formulate the Identity Matrix

The identity matrix \( I \) for a 2x2 matrix is \( \left[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right] \). When multiplied by \( \lambda \), it becomes \( \lambda \cdot I = \left[ \begin{array}{cc} \lambda & 0 \ 0 & \lambda \end{array} \right] \).

Step 3: Subtract \( \lambda I \) from A

Compute \( A - \lambda I \), which is \[ \left[ \begin{array}{cc} a & c \ 0 & b \end{array} \right] - \left[ \begin{array}{cc} \lambda & 0 \ 0 & \lambda \end{array} \right] = \left[ \begin{array}{cc} a - \lambda & c \ 0 & b - \lambda \end{array} \right] \].

Step 4: Solve the Determinant

Calculate the determinant of the matrix \( A - \lambda I \):\[ \text{det}\left( \begin{array}{cc} a - \lambda & c \ 0 & b - \lambda \end{array} \right) = (a - \lambda)(b - \lambda) - (c \cdot 0) = (a - \lambda)(b - \lambda) \].

Step 5: Find the Roots of the Characteristic Equation

Set the determinant from the previous step to zero to find \( \lambda \): \[ (a - \lambda)(b - \lambda) = 0 \]. This gives us two solutions: \( \lambda_1 = a \) and \( \lambda_2 = b \).

Key Concepts

Characteristic Equation

The characteristic equation is a fundamental component in finding eigenvalues of a matrix. It starts with the equation \( \text{det}(A - \lambda I) = 0 \). Here, \( A \) is your given matrix, \( \lambda \) represents the potential eigenvalues, and \( I \) is the identity matrix. To derive this equation, you subtract \( \lambda \) times the identity matrix from \( A \), and then calculate the determinant of this new matrix.

Understanding how we derive the characteristic equation is crucial because eigenvalues are simply the solutions to this equation. By setting \( \text{det}(A - \lambda I) \) equal to zero, you effectively find the values of \( \lambda \) that make the transformation through the matrix \( A \) only involve a stretch or compression, without any rotation. Thus, evaluative insight into matrix transformations can be gained via the characteristic equation.

Identity Matrix

An identity matrix acts as the multiplicative identity in the matrix world, much like how 1 acts in the world of numbers. For a 2x2 identity matrix, it is represented as:

• \( \left[ \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right] \)

When multiplied by any 2x2 matrix, it leaves the original matrix unchanged. In the context of finding eigenvalues, the identity matrix is crucial because it ensures that \( \lambda \) is subtracted uniformly from each diagonal entry of \( A \).

Therefore, multiplying the identity matrix by \( \lambda \) effectively creates a new diagonal matrix \( \lambda I \), which is then subtracted from the original matrix \( A \) to form \( A - \lambda I \). This transformation is key to formulating the characteristic equation, from which eigenvalues can be deduced.

Determinant

The determinant is a special number that can be calculated from a square matrix. It provides important properties, such as invertibility. For a 2x2 matrix, the determinant is calculated as:

• \( \text{det} \left( \begin{array}{cc} p & q \ r & s \end{array} \right) = ps - qr \)

This calculation is particularly crucial when dealing with the matrix \( A - \lambda I \) in the characteristic equation process.

In our specific example, after creating \( A - \lambda I \), the determinant comes out to be \( (a - \lambda)(b - \lambda) \) since the off-diagonal multiplication results in zero, given that one of these off-diagonal elements is 0. The characteristic equation \( \text{det}(A - \lambda I) = 0 \) thus simplifies down to finding when \( (a - \lambda)(b - \lambda) = 0 \), leading us to our eigenvalues \( \lambda_1 = a \) and \( \lambda_2 = b \).

Knowing how to find and use the determinant is fundamental to uncovering vital characteristics of the matrix, such as its eigenvalues in this context.