To find the eigenvalues of a characteristic equation, solve for the values of λ that satisfy det(A - λI) = 0, where A is the matrix and I is the identity matrix.
To find the eigenvalues of a matrix, one typically starts with the characteristic equation, which is obtained by setting the determinant of the matrix subtracted by λ times the identity matrix (A - λI) equal to zero. Mathematically, this is expressed as det(A - λI) = 0. Here, A represents the given matrix, λ denotes the eigenvalue, and I is the identity matrix.
Solving this determinant equation involves finding the values of λ that make the determinant equal to zero. The characteristic equation captures the relationship between the matrix, its eigenvalues, and the identity matrix. The solutions to this equation represent the eigenvalues of the original matrix, providing critical information about the scaling factors of the corresponding eigenvectors. In practical terms, eigenvalues are essential in various fields, including physics, engineering, and data analysis, where they play a crucial role in diagonalizing matrices and understanding linear transformations.