effectiveFisher.py 文件源码

def eigensystem(matrix):
    """
    Given an array-like 'matrix', returns:
        - An array of eigenvalues
        - An array of eigenvectors
        - A rotation matrix that rotates the eigenbasis
            into the original basis

    Example:
        mat = [[1,2,3],[2,4,5],[3,5,6]]
        evals, evecs, rot = eigensystem(mat)
        evals
            array([ 11.34481428+0.j,  -0.51572947+0.j,   0.17091519+0.j]
        evecs
            array([[-0.32798528, -0.59100905, -0.73697623],
                   [-0.73697623, -0.32798528,  0.59100905],
                   [ 0.59100905, -0.73697623,  0.32798528]])
        rot
            array([[-0.32798528, -0.73697623,  0.59100905],
                   [-0.59100905, -0.32798528, -0.73697623],
                   [-0.73697623,  0.59100905,  0.32798528]]))

    This allows you to translate between original and eigenbases:

        If [v1, v2, v3] are the components of a vector in eigenbasis e1, e2, e3
        Then:
            rot.dot([v1,v2,v3]) = [vx,vy,vz]
        Will give the components in the original basis x, y, z

        If [wx, wy, wz] are the components of a vector in original basis z, y, z
        Then:
            inv(rot).dot([wx,wy,wz]) = [w1,w2,w3]
        Will give the components in the eigenbasis e1, e2, e3

        inv(rot).dot(mat).dot(rot)
            array([[evals[0], 0,        0]
                   [0,        evals[1], 0]
                   [0,        0,        evals[2]]])

    Note: For symmetric input 'matrix', inv(rot) == evecs
    """
    evals, emat = eig(matrix)
    return evals, np.transpose(emat), emat