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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 addition of values of its main diagonal, and is noted Tr(M).

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Trace of a Matrix -

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# Trace of a Matrix

## Rectangular Matrix Trace Calculator NxM

### What is the matrix trace? (Definition)

The trace of a square matrix is the addition of the values on its main diagonal (starting from the top left corner and shifting one space to the right and down).

$$\begin{bmatrix} X & . & . \\ . & X & . \\ . & . & X \end{bmatrix} or \begin{bmatrix} X & . & . \\ . & X & . \end{bmatrix} or \begin{bmatrix} X & . \\ . & X \\ . & . \end{bmatrix}$$

### How to calculate a matrix trace?

To calculate the trace of a square matrix $M$ of size $n$, make the sum of diagonal values:

$$\mathrm{Tr}(M) = \sum_{i=1}^{n} a_{i \, i}$$

— For a 2x2 matrix: $$M = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \\ \mathrm{Tr}(M) = a+d$$

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

— 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 $M$ of size $m \times n$, the diagonal used is the one of the included square matrix (from top left corner).

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

### How to find a matrix trace from its eigenvalues?

Calculation from the eigenvalues of a matrix: the trace of a $M$ matrix is equal to the sum of its eigenvalues (including complex values and multiplicity).

NB: The product of the eigenvalues is the determinant of the matrix.

### 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 and B of compatible size (and therefore A.B is a square matrix by multiplication">matrix multiplication):

$$\mathrm{Tr}(AB) = \mathrm{Tr}(BA)$$

For a given scalar c:

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

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

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

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