# Definition: Transposed Matrix

Let $$A\in M_{m\times n}(F)$$ be a matrix with $$m$$ rows and $$n$$ columns and the elements $$\alpha_{ij}$$, $$i=1,\ldots,m$$, $$j=1,\ldots,n$$.

$A=\pmatrix{ \alpha_{11} & \alpha_{12} & \alpha_{13} & \cdots & \alpha_{1n} \\ \alpha_{21} & \alpha_{22} & \alpha_{23} & \cdots & \alpha_{2n} \\ \alpha_{31} & \alpha_{32} & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & \ddots & \alpha_{n-1,n} \\ \alpha_{m1} & \alpha_{m2} & \ldots & \alpha_{m,n-1} & \alpha_{mn}}$

The transposed matrix $$A^T\in M_{n\times m}(F)$$ is a matrix with $$n$$ rows and $$m$$ columns with the elements $$\alpha_{ji}$$, $$i=1,\ldots,m$$, $$j=1,\ldots,n$$:

$A^T=\pmatrix{ \alpha_{11} & \alpha_{21} & \alpha_{31} & \cdots & \alpha_{m1} \\ \alpha_{12} & \alpha_{22} & \alpha_{32} & \cdots & \alpha_{m2} \\ \alpha_{13} & \alpha_{23} & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & \ddots & \alpha_{m,n-1} \\ \alpha_{1n} & \alpha_{2n} & \ldots & \alpha_{n-1,n} & \alpha_{mn}}$

Definitions: 1 2
Explanations: 3

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### References

#### Bibliography

1. Wille, D; Holz, M: "Repetitorium der Linearen Algebra", Binomi Verlag, 1994