Final answer:
The spectral representation of a matrix involves eigenvalues and eigenvectors, and helps understand the properties of the matrix. To determine the spectral representation of matrix A, find its eigenvalues using the characteristic equation and eigenvectors by solving an equation. The spectral representation is given by A = PDP^-1, with D being a diagonal matrix of eigenvalues and P being a matrix of eigenvectors.
Step-by-step explanation:
The spectral representation of a matrix is a representation of the matrix in terms of its eigenvalues and eigenvectors. It helps us understand the behavior and properties of the matrix. To determine the spectral representation of a matrix A, we need to find its eigenvalues and eigenvectors.
Given a matrix A, we start by finding its eigenvalues. The eigenvalues are the roots of the characteristic equation: det(A - λI) = 0, where λ is the eigenvalue and I is the identity matrix. Once we have the eigenvalues, we can find the corresponding eigenvectors by solving the equation (A - λI)x = 0, where x is the eigenvector.
The spectral representation of matrix A is then given by A = PDP^-1, where D is a diagonal matrix with the eigenvalues of A on the diagonal, and P is a matrix whose columns are the corresponding eigenvectors.