Machine Learning Notes - Trace Operator (Trace Operator)

1. What is the trace of a matrix?

        The trace is the sum of all values ​​on the diagonal of a square matrix. Principal Component Analysis (PCA) will use it.

trace of the matrix

        Below is such a matrix A. Then its trace is 2+7+5=14.

        Numpy provides the function trace() to calculate it:

A = np.array([[2, 9, 8], [4, 7, 1], [8, 2, 5]])
A_tr = np.trace(A)

        Get 14.

2. Properties of L2 norm and trace

        Specifies the Frobenius norm of the matrix. The Frobenius norm is equivalent to the L2 norm of a matrix (discussed in the previous chapter).

        It is defined as:

        It can also be calculated by this formula:

        test with python

np.linalg.norm(A)

        You get 17.549928774784245, which means that the L2 norm of A is 17.549928774784245.

        Use traces to calculate

np.sqrt(np.trace(A.dot(A.T)))

        The result is 17.549928774784245.

        Property 1: Since the transpose of a matrix does not change the diagonal, the trace of a matrix is ​​equal to the trace of its transpose:

        Property 2: 

3. Examples

        There are the following three matrices

        

         Calculate the dot product of three matrices

A = np.array([[4, 12], [7, 6]])
B = np.array([[1, -3], [4, 3]])
C = np.array([[6, 6], [2, 5]])

np.trace(A.dot(B).dot(C))

np.trace(C.dot(A).dot(B))

np.trace(B.dot(C).dot(A))

         

        All traces are 531.