The sum of the elements of the principal or main diagonal elements of a square matrix is known as the trace of a matrix. It is generally denoted by Tr(P), where P is any square matrix.
Trace of a matrix example
Let C is a 2 x 2 matrix such that
Now, to find the trace of this matrix C, we add all the elements of the main diagonal elements i.e.
Tr(C) = 3 + 7 = 10
Let D is a 3 x 3 matrix such that
In order to find the trace of matrix D, we add all the elements of the main diagonal elements i.e.
Tr(D) = -1 + 8 + 6 = 13
Trace of matrix E,
Tr(E) = 7 – 5 + 10 – 15 = -3
Trace of a matrix properties
Let P and Q be two square matrices of same order.
1. Tr(P + Q) = Tr(P) + Tr(Q)
2. Tr(PQ) = Tr(QP)
It is true only if both PQ and QP are defined.
3. Tr(kP) = k Tr(P)
where, k is a scalar.