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

Eigenvalues of a matrix \( A \) are found by solving the characteristic equation: \( \det(A - \lambda I) = 0 \), where \( I \) is the identity matrix.

Step 2: Substitute the Matrix into the Equation

Substitute the given matrix \( A = \left[\begin{array}{ll} a & c \ 0 & b \end{array}\right] \) into the eigenvalue equation \( A - \lambda I \). This results in \( \left[\begin{array}{ll} a-\lambda & c \ 0 & b-\lambda \end{array}\right] \).

Step 3: Compute the Determinant

Calculate the determinant of the matrix \( \left[\begin{array}{ll} a-\lambda & c \ 0 & b-\lambda \end{array}\right] \). \( \det = (a-\lambda)(b-\lambda) - 0 \times c = (a-\lambda)(b-\lambda) \).

Step 4: Solve the Characteristic Equation

Solve the equation \( (a-\lambda)(b-\lambda) = 0 \). This implies \( a-\lambda = 0 \) or \( b-\lambda = 0 \).

Step 5: Find the Eigenvalues

Solving each equation, we find \( \lambda_1 = a \) from \( a-\lambda = 0 \), and \( \lambda_2 = b \) from \( b-\lambda = 0 \). Thus, the eigenvalues are \( \lambda_1 = a \) and \( \lambda_2 = b \).

Key Concepts

Eigenvectors

Eigenvectors are special vectors that tell us how a linear transformation behaves. When you have a matrix and multiply it by one of its eigenvectors, you don't get just any vector; you get the same eigenvector back, just scaled by a value known as the eigenvalue. This property makes eigenvectors incredibly helpful for analyzing matrix transformations. For a matrix \( A \), if \( \mathbf{v} \) is an eigenvector corresponding to an eigenvalue \( \lambda \), then the equation holds: \[ A\mathbf{v} = \lambda\mathbf{v} \]Here's why eigenvectors matter:

• They offer a way to decompose matrices, simplifying complex transformations.
• They are used in numerous applications, such as facial recognition and physics simulations.

Understanding eigenvectors is crucial for appreciating the inner workings of matrices, especially in fields that analyze large datasets or apply transformations.

Characteristic Equation

The characteristic equation is how we find the eigenvalues of a matrix. It involves creating a polynomial equation derived from the matrix itself. For a matrix \( A \), the characteristic equation is derived from: \\[ \det(A - \lambda I) = 0 \]This formula subtracts an unknown value \( \lambda \) (which we want to find) multiplied by the identity matrix \( I \). Essentially, you adjust the values along the diagonal, aiming to make the determinant of the resulting matrix zero. The steps to find the characteristic equation typically are:

• Subtract \( \lambda \) from each diagonal element of the original matrix.
• Calculate the determinant of this new matrix.
• Set the determinant to zero and solve for \( \lambda \).

Using the characteristic equation, you can unlock key insights into matrix operations and eigenvalues.

Matrix Determinant

The determinant of a matrix is a scalar value that provides useful information about the matrix, such as whether it is invertible or singular. With square matrices (like the 2x2 matrix in the example), the determinant can quickly indicate if solutions to equations involving the matrix are unique.For a 2x2 matrix \( \left[ \begin{array}{ll} a & b \ c & d \end{array} \right] \), the determinant is calculated as:\[ \det = ad - bc \]Why are determinants important?

• They help in testing invertibility; a matrix with a zero determinant is not invertible.
• The determinant provides information about the matrix's scaling properties in transformations.
• It also plays a crucial role when finding the characteristic equation.

In general, knowing how to find and interpret the determinant allows deeper understanding of both the properties of matrices and the behavior of the systems they represent.