Matrix Transposes

The transpose operation is one that allows us to swap the rows and columns of a matrix. For example, suppose we want to transpose \begin{bmatrix} 4 & 5 & 7 \\ 2 & 3 & 4 \end{bmatrix}. We would change the order of the row and column index, giving us \begin{bmatrix} 4 & 2 \\ 5 & 3 \\ 7 & 4 \end{bmatrix}

Formally, we define the transpose operator as follows. If you have a matrix A, the transpose, A^t, is defined as: (A^T)_{ij} = (A)_{ji}

There are a few properties of transposing matrices that are helpful to know. If A and B are matrices and s is a real number scalar, then:

  1. (A^t)^T = A
  2. (A+B)^T = A^T + B^T
  3. (sA)^T = sA^T

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