Problem 57 - Calculus for Biology and Medicine

Question

In Problems \(57-60\), find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for each matrix \(A\). $$A=\left[\begin{array}{ll}4 & 0 \\ 0 & 3\end{array}\right]$$

Step-by-Step Solution

Answer

The eigenvalues are \(\lambda_1 = 4\) and \(\lambda_2 = 3\).

Step 1: Eigenvalue Definition

To find the eigenvalues of matrix \(A\), we use the equation \(A\mathbf{v} = \lambda\mathbf{v}\), where \(\lambda\) is a scalar called the eigenvalue, and \(\mathbf{v}\) is the eigenvector.

Step 2: Characteristic Equation

The characteristic equation is obtained from \(\det(A - \lambda I) = 0\), where \(I\) is the identity matrix of the same size as \(A\).

Step 3: Substitute the Matrices

Substitute \(A\) and \(I\) into the characteristic equation: \(\det\left(\begin{bmatrix} 4 & 0 \ 0 & 3 \end{bmatrix} - \lambda \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\right) = 0\).

Step 4: Simplify the Matrix Expression

Simplifying the expression inside the determinant gives \(\begin{bmatrix} 4 - \lambda & 0 \ 0 & 3 - \lambda \end{bmatrix}\).

Step 5: Calculate the Determinant

The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is given by \(ad - bc\). Thus, calculate \((4 - \lambda)(3 - \lambda) - 0 = 0 \).

Step 6: Solve the Quadratic Equation

Expand and solve the equation \((4 - \lambda)(3 - \lambda) = 0\). This simplifies to \(\lambda^2 - 7\lambda + 12 = 0\).

Step 7: Factor and Find Eigenvalues

The equation \(\lambda^2 - 7\lambda + 12 = 0\) can be factored as \((\lambda - 4)(\lambda - 3) = 0\). Thus, the eigenvalues are \(\lambda_1 = 4\) and \(\lambda_2 = 3\).

Key Concepts

Eigenvectors

Eigenvectors are an important concept when dealing with matrices. They provide us with insight into the geometric properties of the matrix. An eigenvector, denoted as \(\mathbf{v}\), is a non-zero vector that changes at most by a scalar factor when that linear transformation is applied. This is handy when you want to understand the essential directions a transformation affects.
To find the eigenvectors, we have to use the equation \(A\mathbf{v} = \lambda\mathbf{v}\). Here, \(A\) is the matrix, \(\mathbf{v}\) is the eigenvector, and \(\lambda\) is the eigenvalue. The key idea is that when matrix \(A\) acts on \(\mathbf{v}\), the output is the same vector scaled by \(\lambda\).

• Eigenvectors show in which directions transformation due to a matrix stretches or compresses space.
• They provide directions along which the linear transformation specified by \(A\) acts.

To determine eigenvectors, one would typically solve the equation \((A - \lambda I)\mathbf{v} = 0\). This involves inserting each eigenvalue back into the matrix and solving the homogeneous equation for the vector \(\mathbf{v}\). This task might seem complex initially, but with practice, you can easily determine these vectors.

Characteristic Equation

The characteristic equation is central to finding the eigenvalues of a matrix. It offers a path to uncovering these values by setting up an equation derived from the matrix itself.
The characteristic equation is formed using the determinant. Specifically, it is \(\det(A - \lambda I) = 0\), where \(A\) is the matrix you're investigating and \(I\) is the identity matrix. This equation sets the stage to solve for \(\lambda\), the eigenvalues.

• The process involves subtracting \(\lambda\) times the identity matrix \(I\) from the matrix \(A\).
• The determinant of this new matrix gives us the characteristic polynomial.
• Solving this polynomial for zero provides the eigenvalues of the original matrix \(A\).

Through the characteristic equation, we determine the scalar values that characterize the transformation properties of the matrix. These values, the eigenvalues, reveal the scale of transformation along each eigenvector, providing crucial insights into the matrix's behavior.

Determinant

The determinant is a special value that can be calculated from a square matrix. It plays a pivotal role in linear algebra, particularly when finding the eigenvalues of a matrix.
In the context of finding eigenvalues, we often calculate the determinant of the matrix \((A - \lambda I)\), where \(A\) is your original matrix, and \(I\) is the identity matrix of the same size.

• The determinant offers a single number summarizing certain properties of the matrix.
• For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated as \(ad - bc\).
• A non-zero determinant indicates that a matrix has full rank and is invertible.

When solving for eigenvalues using the determinant in the characteristic equation, we're specifically trying to find where this determinant equals zero. This zero mark leads to the characteristic polynomial from which eigenvalues are derived. Understanding the determinant's role is key in matrix algebra, as it gives insight into solvability and transformations represented by the matrix.