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Since a digital image can be considered as a matrix, we may ask how the operations with its elements influence the corresponding image. For example, given the matrix $A = (a_{i,j}),$ we may consider the matrix $B = (b_{i,j}) = (a_{j,i})},$ the transposed matrix of $A,$ obtained replacing the rows of $A$ by its columns. For example, $$ \mbox{if } \quad A = \left[\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right], \quad \mbox{ then } \quad B = A^{T} = \left[\begin{array}{cc} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{array}\right], $$ since $$ \begin{array}{ccccccccccc} b_{1,1} & = & a_{1,1} & = & 1, & \qquad & b_{1,2} & = & a_{2,1} & = & 4, \\ b_{2,1} & = & a_{1,2} & = & 2, & & b_{2,2} & = & a_{2,2} & = & 5, \\ b_{3,1} & = & a_{1,3} & = & 3, & & b_{3,2} & = & a_{2,3} & = & 6. \end{array} $$ |
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