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Trace of a Matrix

Tool to compute the trace of a matrix. The trace of a square matrix M is the sum of its main diagonal denoted Tr(M).

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Trace of a Matrix -

Tag(s) : Mathematics,Algebra,Symbolic Computation

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Trace of a Matrix

Rectangular Matrix Trace Calculator NxM

Tool to compute the trace of a matrix. The trace of a square matrix M is the sum of its main diagonal denoted Tr(M).

How to calculate a matrix trace?

For a square matrix M of size n, you make the sum of diagonal values:$$\mathrm{Tr}(M) = \sum_{i=1}^{n} a_{i \, i}$$For rectangular matrix, the diagonal used is the one of the included square matrix.

For a 2x2 matrix : $$M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$

$${\rm Tr}(M) = a+d$$

For a 3x3 matrix : $$M = \begin{pmatrix} a & b & c \\d & e & f \\ g & h & i \end{pmatrix}$$

$${\rm Tr}(M) = a+e+i$$

What are trace mathematical properties?

Trace follows the following properties :

For A and B of the same order (that can be added):

$$\rm{Tr}(A + B) = \rm{Tr}(A) + \rm{Tr}(B)$$

For a given scalar c:

$$\rm{Tr}(c A) = c \rm{Tr}(A)$$

For $$A^T$$ the transposed matrix of A:

$$\rm{Tr}(A^T) = \rm{Tr}(A)$$