Problem 58 - Calculus for Biology and Medicine

Question

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for each matrix \(A\). $$A=\left[\begin{array}{rr}-7 & 0 \\ 0 & 6\end{array}\right]$$

Step-by-Step Solution

Answer

The eigenvalues are \( \lambda_1 = -7 \) and \( \lambda_2 = 6 \).

Step 1: Identify the Characteristic Equation

The characteristic equation of a matrix \( A \) is given by \( \det(A - \lambda I) = 0 \), where \( I \) is the identity matrix of the same size as \( A \) and \( \lambda \) represents the eigenvalues.

Step 2: Substitute the Matrix Values

For the matrix \( A = \begin{bmatrix} -7 & 0 \ 0 & 6 \end{bmatrix} \), the identity matrix \( I \) is \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \). Then the matrix \( A - \lambda I \) is \( \begin{bmatrix} -7 - \lambda & 0 \ 0 & 6 - \lambda \end{bmatrix} \).

Step 3: Compute the Determinant

The determinant of \( A - \lambda I \) is \( (-7 - \lambda)(6 - \lambda) \). Set this equal to zero to find the characteristic equation: \((-7 - \lambda)(6 - \lambda) = 0 \).

Step 4: Solve the Characteristic Equation

Solve the equation \((-7 - \lambda)(6 - \lambda) = 0 \) by setting each factor equal to zero: - For \(-7 - \lambda = 0 \), we find \( \lambda = -7 \).- For \(6 - \lambda = 0 \), we find \( \lambda = 6 \).

Step 5: Identify the Eigenvalues

The solutions \( \lambda = -7 \) and \( \lambda = 6 \) are the eigenvalues of the matrix \( A \). Thus, the eigenvalues are \( \lambda_1 = -7 \), \( \lambda_2 = 6 \).

Key Concepts

Characteristic Equation

The characteristic equation is fundamental in finding the eigenvalues of a matrix. It is formulated as \( \det(A - \lambda I) = 0 \). Here, \( A \) represents the original matrix, and \( \lambda \) signifies the eigenvalues we aim to find. The term \( I \) stands for the identity matrix, which is crucial as it serves as a neutral element in matrix operations. Think of this equation as a special form where we "subtract" a multiple of the identity matrix from \( A \) to make a determinant, which provides the condition for certain eigenvalues of \( A \).

• Find \( A - \lambda I \) by subtracting \( \lambda \) times the identity matrix from \( A \).
• Compute the determinant of the result to form a polynomial equation.
• The solutions to this polynomial are the eigenvalues.

Understanding this equation helps in solving for eigenvalues, which are crucial in many applications of linear algebra.

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. It's an essential concept in linear algebra because it sums up key properties of the matrix, such as invertibility. When working with eigenvalues, the determinant indicates whether a matrix can be simplified to focus on easily solvable linear forms.

• In our context, it's used to solve the characteristic equation, formulated as \( \det(A - \lambda I) = 0 \).
• A determinant equal to zero in this equation implies the equation has solutions for eigenvalues.

For a 2x2 matrix, such as the one given in the exercise, the determinant is straightforward to calculate as the product of the diagonal elements minus the product of the off-diagonal elements. Ease with determinants makes it simpler to find eigenvalues.

Matrix Subtraction

Matrix subtraction occurs when you subtract one matrix from another. In the case of eigenvalues, matrix subtraction involves subtracting \( \lambda I \) from \( A \), where \( \lambda \) is an unknown scalar and \( I \) is the identity matrix. This operation is integral to forming the characteristic equation.

• The subtraction \( A - \lambda I \) modifies the matrix \( A \) to account for the unknown \( \lambda \).
• The resulting matrix is \( A \) with \( \lambda \) subtracted from its diagonal elements.

This step simplifies the matrix to help evaluate its determinant and solve for \( \lambda \). Matrix subtraction is similar to standard arithmetic, with easy-to-follow rules, yet it takes on a pivotal role in finding eigenvalues.

Identity Matrix

An identity matrix is a square matrix with 1s on the diagonal and 0s elsewhere. It acts as the "1" element of matrix multiplication, maintaining the original matrix untouched when multiplied by it. The identity matrix is key when forming the expression \( A - \lambda I \) in the characteristic equation.

• The identity matrix ensures that \( \lambda \) is only subtracted from the diagonal elements of \( A \).
• It preserves the structure of the matrix, allowing us to isolate changes due to \( \lambda \).

For the given matrix \( A = \begin{bmatrix} -7 & 0 \ 0 & 6 \end{bmatrix} \), the identity matrix is \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \). This simplicity keeps algebraic manipulations straightforward and underpins much of the eigenvalue calculation process.