Eigenvalues II, Matrix Diagonalization and Unitary Matrix

In this chapter, we will show another method to find the eigenvalues of a matrix. We further show that the process of finding the eigenvalues of a matrix is equivalent to diagonalizing the matrix. Matrix diagonalization means to represent the matrix in its eigenvector basis and thus it is diagonal. We also discuss another important type of matrix, namely the unitary matrix. All quantum gates need to be unitary because they preserve the inner products of the vectors when the vectors are transformed/rotated by the unitary matrix.