Matrix Trace Operations

The matrix (phalanx) trace operation returns the and of the diagonal elements of the matrix:

Trace operations are useful for a number of reasons. With the summation notation, some matrix operations are difficult to describe, and the matrix multiplication and trace operation symbols can be clearly expressed. For example Matrix F-norm (Frobenius norm):

Trace Operation Property 1:

Set a square a, have

Trace Operation Property 2:

Multiple matrices are multiplied by the traces of the square matrix, and the traces of the last one of these matrices are multiplied before being moved to the front. Of course, we need to consider moving after the matrix product is still well-defined:

For example, suppose that matrix A is a m*n matrix andB is a n*m matrix, then:tr (AB) = TR (BA). You can see that although AB is a m*m matrix and BA is a n*n matrix, the result of the trace operation is equal.

Trace Operation Property 3:

The trace operation of a scalar is its own,Tr (a) = a

Trace Operation Property 4: Similarity invariance of Traces

If matrices A and B are similar, they will have the same trace, and the definition of the similarity of matrices will not be said.