Finding eigenvalues of a matrix involves determining the values of \(\lambda\) that satisfy the **characteristic equation** of the matrix. Here are the detailed steps to find eigenvalues:

### Step 1: **Define the matrix**
Let \(A\) be an \(n \times n\) matrix. The goal is to find eigenvalues \(\lambda\) such that there exists a non-zero vector \(v\) (eigenvector) that satisfies the equation:

\[
A v = \lambda v
\]

### Step 2: **Set up the characteristic equation**
Rearranging the equation \(A v = \lambda v\), we get:

\[
A v - \lambda I v = 0
\]

where \(I\) is the identity matrix of the same dimension as \(A\). This can be rewritten as:

\[
(A - \lambda I) v = 0
\]

For non-zero eigenvectors \(v\), this equation only has non-trivial solutions if the matrix \((A - \lambda I)\) is **singular**. The matrix is singular if and only if its determinant is zero, so we get the **characteristic equation**:

\[
\det(A - \lambda I) = 0
\]

### Step 3: **Compute the determinant**
Calculate the determinant of the matrix \(A - \lambda I\). This will result in a polynomial in terms of \(\lambda\), known as the **characteristic polynomial**.

For example, if \(A\) is a \(2 \times 2\) matrix:

\[
A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
\]

then the characteristic equation is:

\[
\det \begin{pmatrix} a - \lambda & b \\ c & d - \lambda \end{pmatrix} = (a - \lambda)(d - \lambda) - bc = 0
\]

This simplifies to a quadratic equation in \(\lambda\):

\[
\lambda^2 - (a+d)\lambda + (ad - bc) = 0
\]

### Step 4: **Solve the characteristic polynomial**
Solve the characteristic polynomial for \(\lambda\). The roots of this polynomial are the **eigenvalues** of the matrix \(A\).

For the \(2 \times 2\) example above, solve the quadratic equation:

\[
\lambda^2 - (a+d)\lambda + (ad - bc) = 0
\]

using the quadratic formula:

\[
\lambda = \frac{-(a+d) \pm \sqrt{(a+d)^2 - 4(ad - bc)}}{2}
\]

### Step 5: **Verify the results**
After solving for the eigenvalues, verify by plugging them back into the matrix equation \(A v = \lambda v\) to find the corresponding eigenvectors, or use numerical methods for large matrices.

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Would you like a specific example to clarify this, or do you have any questions?

### Relative Questions:
1. How do you find the eigenvectors after determining the eigenvalues?
2. What is the significance of eigenvalues in real-world applications?
3. Can there be complex eigenvalues, and how are they interpreted?
4. How does the process change for larger matrices (e.g., \(3 \times 3\))?
5. What are the practical computational methods for finding eigenvalues in large matrices?

### Tip:
Eigenvalues help to understand matrix behavior in linear transformations, such as scaling, rotating, and shearing of vectors.

Learn how to find the eigenvalues of a matrix with a detailed, step-by-step guide. This tutorial covers key linear algebra concepts, including the eigenvalue theorem, characteristic equation, and the quadratic formula. Suitable for undergraduate students.

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