Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

G.7 PROPERTIES OF TRACE AND DETERMINANT

Definition: Trace The trace of square matrix is

(G.53)

where {ann} are the diagonal elements of A.

The trace has the following properties:

• Eigenvalues :

  (G.54)

• Addition and multiplication by scalars {c, d}:

  (G.55)

• Associative pairwise: tr[AB] = tr[BA] and

  (G.56)

  (G.57)

• Kronecker product:

  (G.58)

• Similarity invariant:

  (G.59)

  for invertible matrix P.
• Bounds for PSD matrices A and B:

  (G.60)

The determinant of square matrix has the following ...