### Eigenvalues & Eigenvectors

Calculating **eigenvalues and eigenvectors of a square matrix** A involves finding solutions to the equation \(Ax = \lambda x\), where \(\lambda\) is the eigenvalue and \(x\) is the eigenvector. Here are the steps for finding eigenvalues and eigenvectors:

- 1.Calculate the determinant of the matrix \(A - \lambda I\), where I is the identity matrix. The determinant of \(A - \lambda I\) is known as the
**characteristic polynomial**of the matrix A. - 2.Set the determinant equal to zero and solve for \(\lambda\). The values of \(\lambda\) that satisfy this equation are the eigenvalues of the matrix A.
- 3.For each eigenvalue \(\lambda\), solve the equation \((A - \lambda I)x = 0\) to find the corresponding eigenvectors. This is equivalent to finding the null space of the matrix \(A - \lambda I\).
- 4.Normalize the eigenvectors by dividing them by their length to obtain the unit eigenvectors.

Alternatively, there are different methods for finding eigenvalues and eigenvectors, including:

- 1.
**The power method**: This method involves multiplying the matrix A repeatedly by a random vector until it converges to an eigenvector. The eigenvalue can be calculated using the Rayleigh quotient. - 2.
**QR algorithm**: This method involves applying a series of QR decompositions to the matrix A until it becomes diagonal, with the eigenvalues on the diagonal. The eigenvectors can then be obtained by back-substitution. - 3.
**Jacobi method**: This method involves applying a series of orthogonal transformations to the matrix A until it becomes diagonal, with the eigenvalues on the diagonal. The eigenvectors can then be obtained by extracting the columns of the orthogonal matrix.

It's important to note that not all matrices have eigenvalues and eigenvectors, and some matrices have repeated eigenvalues or non-real eigenvalues. Additionally, the eigenvalues and eigenvectors of a matrix may not always be easy to calculate analytically, especially for large matrices. In these cases, numerical methods may be used to approximate the eigenvalues and eigenvectors.