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.

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