Transposed matrix

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The matrix obtained from a given (rectangular or square) matrix $A=\|a_{ik}\|$ ($i=1,\dots,m$; $k=1,\dots,n$) by interchanging the rows and the columns, that is, the matrix $\|a_{ik}'\|$, where $a_{ik}'=a_{ki}$ ($i=1,\dots,n$; $k=1,\dots,m$). The number of rows of the transposed matrix is equal to the number of columns of $A$, while the number of columns is equal to the number of rows of $A$. The transpose of a matrix $A$ is usually denoted by $A^T$ or $A'$.


Some elementary properties of the transposition of matrices are $(A+B)^T=A^T+B^T$, $(\alpha A)^T=\alpha A^T$, $(AB)^T=B^TA^T$, $(A^{-1})^T=(A^T)^{-1}$.


[a1] F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1959) pp. 19 (Translated from Russian)
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