To find eigenvalues and eigenvectors of a square matrix, you need to follow these steps:

Steps to Find Eigenvalues:

1. Start with a square matrix $A$: The matrix must be square (e.g., $2 \times 2$, $3 \times 3$) for eigenvalues and eigenvectors to exist.

2. Set up the characteristic equation: The characteristic equation is given by:

   $\text{det}(A - \lambda I) = 0$

   Here:

   • $A$ is your original matrix.
   • $\lambda$ is the eigenvalue.
   • $I$ is the identity matrix of the same size as $A$.

3. Solve for $\lambda$ (the eigenvalues): Find the values of $\lambda$ that satisfy the characteristic equation. These are the eigenvalues.

Steps to Find Eigenvectors:

1. Substitute each eigenvalue $\lambda$ into the equation:

   $(A - \lambda I) \mathbf{v} = 0$

   Here:

   • $\mathbf{v}$ is the eigenvector corresponding to the eigenvalue $\lambda$.

2. Solve the system of linear equations: For each $\lambda$, solve the matrix equation $(A - \lambda I) \mathbf{v} = 0$ to find the corresponding eigenvector $\mathbf{v}$. You will usually find the eigenvector up to a constant multiple, as any scalar multiple of an eigenvector is also an eigenvector.

Example:

Let's find the eigenvalues and eigenvectors for a simple $2 \times 2$ matrix:

4 & 1 \\ 2 & 3 \end{pmatrix}$$ 1. Set up the characteristic equation: $$\text{det}(A - \lambda I) = 0$$ $$\text{det}\left(\begin{pmatrix} 4 - \lambda & 1 \\ 2 & 3 - \lambda \end{pmatrix}\right) = 0$$ 2. Compute the determinant: $$(4 - \lambda)(3 - \lambda) - (1 \cdot 2) = 0$$ $$\lambda^2 - 7\lambda + 10 = 0$$ Solve this quadratic equation: $$\lambda = 5 \quad \text{or} \quad \lambda = 2$$ 3. For each eigenvalue, solve for the eigenvectors. Let's start with $$\lambda = 5$$: $$(A - 5I) \mathbf{v} = 0$$ $$\begin{pmatrix} -1 & 1 \\ 2 & -2 \end{pmatrix} \mathbf{v} = 0$$ Solve this system, and you'll find $$\mathbf{v} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$$. This is an eigenvector corresponding to $$\lambda = 5$$. 4. Similarly, solve for $$\lambda = 2$$ to find another eigenvector. Would you like a more detailed explanation or another example? ### Related Questions: 1. What are the properties of eigenvalues and eigenvectors? 2. How are eigenvalues related to the trace and determinant of a matrix? 3. Can a matrix have complex eigenvalues? 4. How do you find eigenvalues for larger matrices (e.g., $$3 \times 3$$)? 5. What are the geometric interpretations of eigenvalues and eigenvectors? ### Tip: For larger matrices or more complicated calculations, you can use software tools like MATLAB, NumPy (Python), or even graphing calculators to compute eigenvalues and eigenvectors efficiently.