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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?

To calculate the trace of 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 a 2x2 matrix : $$M = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \\ \mathrm{Tr}(M) = a+d$$

$$M = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \\ \mathrm{Tr}(M) = 1+4 = 5$$

For a 3x3 matrix : $$M = \begin{bmatrix} a & b & c \\d & e & f \\ g & h & i \end{bmatrix} \\ \mathrm{Tr}(M) = a+e+i$$

For rectangular matrix, the diagonal used is the one of the included square matrix (from top left corner).

$$M = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \Rightarrow \mathrm{Tr}(M) = \mathrm{Tr} \begin{bmatrix} 1 & 2 \\ 4 & 5 \end{bmatrix}$$

### What are trace mathematical properties?

Trace follows the following properties:

The trace of an identity matrix $$I_n$$ of size $$n$$ equals $$n$$.

$$\mathrm{Tr}(I_n) = n$$

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

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

For a given scalar c:

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

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

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