short method to find eigen value of symmetric matrix

To find the eigenvalues of a symmetric matrix, you can use the following short method:

### Steps:
1. **Write the characteristic equation**: The characteristic equation for a matrix \(A\) is given by:
   \[
   \det(A - \lambda I) = 0
   \]
   where:
   - \(A\) is the symmetric matrix.
   - \(\lambda\) is the eigenvalue.
   - \(I\) is the identity matrix of the same size as \(A\).

2. **Solve the characteristic polynomial**: The determinant \(\det(A - \lambda I)\) will give you a polynomial equation in terms of \(\lambda\). Solve for \(\lambda\), which will give the eigenvalues.

3. **Properties of symmetric matrices**: Symmetric matrices have the following properties:
   - All eigenvalues are real numbers.
   - Eigenvectors corresponding to distinct eigenvalues are orthogonal.

### Short Method for 2x2 Matrices:
For a \(2 \times 2\) symmetric matrix \( A = \begin{pmatrix} a & b \\ b & c \end{pmatrix} \), the characteristic equation simplifies to:
\[
\det \begin{pmatrix} a-\lambda & b \\ b & c-\lambda \end{pmatrix} = 0
\]
This leads to the quadratic equation:
\[
(a-\lambda)(c-\lambda) - b^2 = 0
\]
Simplify and solve for \(\lambda\).

### Example:
For the matrix \( A = \begin{pmatrix} 4 & 1 \\ 1 & 3 \end{pmatrix} \):
1. The characteristic equation is:
   \[
   \det \begin{pmatrix} 4-\lambda & 1 \\ 1 & 3-\lambda \end{pmatrix} = 0
   \]
2. Expanding the determinant gives:
   \[
   (4 - \lambda)(3 - \lambda) - 1^2 = 0
   \]
3. Solving the quadratic equation:
   \[
   \lambda^2 - 7\lambda + 11 = 0
   \]
4. The solutions for \(\lambda\) are the eigenvalues.

### Tip:
For larger matrices, using **matrix diagonalization** (if the matrix is diagonalizable) or **numerical methods** such as **QR algorithm** might be efficient for finding eigenvalues quickly.

Would you like more detailed steps for a specific example or explanation?

Learn how to quickly find the eigenvalues of a symmetric matrix, using the characteristic equation and a simplified method for 2x2 matrices. The guide covers essential concepts in linear algebra, including determinants and eigenvalues, with a detailed example. Ideal for high school students and university beginners in linear algebra.

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