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.