Calculate the trace of a square matrix using the matrix trace calculator. Select the dimension of a matrix among 2x2, 3x3, 4x4, 5x5, 6x6, 7x7, 8x8 and fill the matrix with values. The calculator will calculate the trace of the matrix for you.
The trace of a matrix is the sum of all the elements in its main diagonal.
Only a square matrix has a trace.
For example, there is a matrix A, The trace of a matrix A is denoted by tr(A).
Real world Application of trace of matrix
Trace of a matrix is not just a mathematical concept but it has real world applications too.
Its used in following fields:
1. Quantum Mechanics
2. Computer Graphics
3. Machine Learning
4. Robotics
5. Physics
6. Economics
Properties of trace of matrix
Say, there is a matrix A
1. tr(A) = tr(A^T)
Trace of a matrix is always equal to the trace of its transpose matrix
2. tr(A + B) = tr(A) + tr(B)
The trace of the sum of two matrices is equal to the sum of the traces of the matrices
3. tr(λA) = λ * tr(A)
The trace of a scalar multiple of a matrix is equal to the scalar multiple of the trace of the matrix
4. tr(AB) = tr(BA)
This is the most interesting property of trace of a matrix. The order of multiplication of two matrices doesn't matter. The trace of the resultant matrix is always the same value
And the multiplication property holds true for any number of matrices. tr(PQR) = tr(RPQ) = tr(QRP) = tr(PQR) = tr(RQP) = tr(QPR) = ... = tr(...)
How to calculate the trace of a matrix?
To calculate the trace of a matrix, the matrix must be a square matrix.
