Problem 59 - Calculus for Biology and Medicine

Question

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for each matrix \(A\). $$A=\left[\begin{array}{lr}1 & -3 \\ 0 & 2\end{array}\right]$$

Step-by-Step Solution

Answer

The eigenvalues are \( \lambda_1 = 1 \) and \( \lambda_2 = 2 \).

Step 1: Understand the Eigenvalue Equation

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

Step 2: Set up the Matrix \( A - \lambda I \)

Given the matrix \( A = \begin{bmatrix} 1 & -3 \ 0 & 2 \end{bmatrix} \) and identity matrix \( I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \). Compute \( A - \lambda I = \begin{bmatrix} 1-\lambda & -3 \ 0 & 2-\lambda \end{bmatrix} \).

Step 3: Compute the Determinant

The determinant of \( A - \lambda I \) is given by the formula: \[ \det(A - \lambda I) = (1-\lambda)(2-\lambda) - (0)(-3) = (1-\lambda)(2-\lambda) \]

Step 4: Solve the Characteristic Equation

Set the determinant equal to zero: \((1-\lambda)(2-\lambda) = 0\) Solve this for \( \lambda \): \[ \lambda_1 = 1, \lambda_2 = 2 \]

Step 5: Confirm Eigenvalues

The roots of the characteristic equation give the eigenvalues of the matrix \( A \). Thus, \( \lambda_1 = 1 \) and \( \lambda_2 = 2 \) are the eigenvalues.

Key Concepts

Eigenvectors

Eigenvectors are special vectors associated with a matrix that remain parallel to their original direction after a linear transformation is applied via the matrix. Imagine transforming an object, like rotating or stretching a shape, without changing the direction of some of its axes; those are like the eigenvectors.
These vectors work with eigenvalues, which indicate how much they stretch or compress. Given a matrix, identifying eigenvectors can help analyze its properties like stability and symmetry.

• To find an eigenvector, use the equation \( (A - \lambda I) \mathbf{v} = \mathbf{0} \).
• The vector \( \mathbf{v} \) that satisfies this equation for a corresponding eigenvalue \( \lambda \) is an eigenvector.
• Typically, you need to solve a system of linear equations to determine the eigenvector components.

Understanding eigenvectors gives insights into the dynamics governed by the matrix, aiding in simplifications used in engineering and physics.

Linear Algebra

Linear algebra is a branch of mathematics focused on vectors, vector spaces, and linear transformations. It's essential for understanding a wide range of scientific fields and technologies.
At its core, linear algebra deals with equations of lines and planes, and it extends to higher dimensions with structures like matrices.

• Systems of linear equations are solved using vectors and matrices.
• Operations include addition, subtraction, and multiplication of vectors and matrices.

Understanding concepts like eigenvalues and eigenvectors deepen the application of linear algebra in computational sciences, graphics, and more.
For matrices like the one in the exercise, this field provides the tools to understand complex transformations and properties of space.

Characteristic Equation

The characteristic equation relates to finding eigenvalues of a matrix. It is derived from the equation \( \det(A - \lambda I) = 0 \), where \( A \) is a square matrix and \( \lambda \) represents an eigenvalue.
This equation encapsulates the essence of how a matrix can be expressed via its eigenvalues, which simplifies many computations.

• Each root of the characteristic equation corresponds to an eigenvalue of the matrix.
• Solving \( \det(A - \lambda I) = 0 \) involves determinant calculations, which simplify into a polynomial whose roots are the sought eigenvalues.

The characteristic equation's roots reveal a matrix's structure and behavior, playing a crucial role in mathematical modeling and problem-solving in various fields.