Question

Find the eigenvalues and eigenvectors of the following matrices.\(\left(\begin{array}{lll}3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & 3\end{array}\right)\)

Short answer

Calculate the characteristic polynomial, solve for \(\lambda\) to find eigenvalues, and then solve \(A - \lambda I\) for each eigenvalue to find the eigenvectors.

Step-by-step solution

- Write Down the Characteristic Equation

The eigenvalues of a matrix are found by solving the characteristic equation. The characteristic equation is given by \(\det(A - \lambda I) = 0\), where \(A\) is the matrix and \(\lambda\) is the eigenvalue. For the given matrix \(A = \left(\begin{array}{lll}3 & 2 & 4 \ 2 & 0 & 2 \ 4 & 2 & 3\end{array}\right)\) and the identity matrix \(I\), calculate \(A - \lambda I\).

- Calculate \(A - \lambda I\)

Subtract \(\lambda\) times the identity matrix from \(A\): \[\left( \begin{array}{ccc} 3 & 2 & 4 \ 2 & 0 & 2 \ 4 & 2 & 3 \end{array} \right) - \lambda \left( \begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array} \right) = \left( \begin{array}{ccc} 3-\lambda & 2 & 4 \ 2 & -\lambda & 2 \ 4 & 2 & 3-\lambda \end{array} \right)\].

- Find the Determinant of \(A - \lambda I\)

Calculate the determinant of the matrix \(\left( \begin{array}{ccc} 3-\lambda & 2 & 4 \ 2 & -\lambda & 2 \ 4 & 2 & 3-\lambda \end{array} \right)\). Use the cofactor expansion method to find the determinant: \[\det(A - \lambda I) = \begin{vmatrix} 3-\lambda & 2 & 4 \ 2 & -\lambda & 2 \ 4 & 2 & 3-\lambda \end{vmatrix}= (3-\lambda) \begin{vmatrix} -\lambda & 2 \ 2 & 3-\lambda \end{vmatrix} - 2 \begin{vmatrix} 2 & 2 \ 4 & 3-\lambda \end{vmatrix} + 4 \begin{vmatrix} 2 & -\lambda \ 4 & 2 \end{vmatrix}\].

- Simplify the Determinant

Evaluate each determinant: \[\det(A - \lambda I) = (3-\lambda)[(-\lambda(3-\lambda) - 4)] - 2[(2(3-\lambda) - 8)] + 4[(2(-\lambda) - 8)]\]. Simplify further to get the characteristic polynomial in terms of \(\lambda\).

- Solve the Characteristic Polynomial

Solve the polynomial obtained in Step 4 for \(\lambda\). This will give the eigenvalues.

- Find the Eigenvectors

Substitute each eigenvalue \(\lambda_i\) back into the equation \(A - \lambda_i I\) and solve the resulting system of linear equations for the eigenvectors.

Key concepts

Characteristic Equation

To determine the eigenvalues of a matrix, start with the **characteristic equation**. This equation is derived by subtracting \( \lambda I \) from the matrix A, and then finding the determinant of the resulting matrix. The characteristic equation is given by \[ \det(A - \lambda I) = 0 \]. Here, \( A \) is the matrix in question, \( I \) is the identity matrix of the same dimensions as \( A \), and \( \lambda \) represents the eigenvalue. Solving this determinant equation yields the eigenvalues.

Determinant

Next, we need to find the determinant of the matrix \( A - \lambda I \). Given matrix \( A \), and identity matrix \( I \), first create the matrix \( A - \lambda I \) by subtracting \( \lambda <> I \) from each corresponding element of \( A \). The determinant of the resultant matrix is computed using methods such as cofactor expansion or Laplace's formula. For instance, for a 3x3 matrix, you expand along any row or column. Calculating these determinants step-by-step is crucial:

• Write down the matrix \( A - \lambda I \).
• Perform the row or column expansion to find sub-determinants.
• Simplify to get a polynomial equation in \( \lambda \).

Remember, the solutions to this polynomial are the eigenvalues.

Eigenvector Calculation

Once the eigenvalues \( \lambda_i \) are found, the next task is to determine the corresponding eigenvectors. Insert each eigenvalue \( \lambda_i \) back into the expression \( (A - \lambda_i I) \mathbf{x} = 0 \), where \( \mathbf{x} \) is the eigenvector. This results in a homogeneous system of linear equations. Find this vector by:

• Substituting \( \lambda_i \) into \( A - \lambda_i I \).
• Solving the resulting equation system for the components of \( \mathbf{x} \).

Eigenvectors are typically normalized so that they have a length of 1, making them unique representations within their eigenspace.

Linear Algebra Methods

Linear Algebra provides the tools necessary to solve for eigenvalues and eigenvectors efficiently. Some common methods include:

• **Matrix Decomposition Techniques:** Methods like QR decomposition can simplify the computation of eigenvalues and eigenvectors.
• **Numerical Methods:** Algorithms such as the Power Method, Inverse Iteration, and the QR algorithm can find estimates of eigenvalues and eigenvectors, particularly useful for large matrices.
• **Symbolic Computation:** Use of software (like MATLAB or Mathematica) can perform the algebraic manipulation needed to solve characteristic equations and determinants.

Applying these techniques helps to break down complex problems, making them easier to understand and solve.