Trace Of Matrix

Linear Algebra / By

Matrix has a special function called trace function. If A is a square matrix then the sum of its main diagonal entry is called trace of matrix A and is denoted by tr(A).

Let A be a square matrix with size n\times n, then

A = \begin{bmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{bmatrix}

The trace of matrix is 

tr(A) = a_{11} + b_{22} + c_{33}

Let is see few examples of traces of matrices.

Example #1

Let A be a square matrix of size 3 \times 3

A = \begin{bmatrix}-1 & 4 & 2\\6 & 2 & 7\\5 & 1 & 8\end{bmatrix}

The trace of matrix is 

tr(A) = (-1) + 2 + 8 = 9

tr(A) = 9

Example #2

Let B be a square matrix of size 4 \times 4

B = \begin{bmatrix}6 & 1 & 1 & -2\\5 & 9 & -1 & 3\\0 & 1 & 7 & 2\\3 & 7 & 8 & 5\end{bmatrix}

The trace of matrix is 

tr(B) = 6  + 9 + 7 + 5 = 27

tr(B) = 27

If the matrix A is not a square matrix, then tr(A) is not defined.

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