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I can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the transpose, or even symmetric matrices.

For example, if I have a linear transformation, say on the plane, my intuition is to visualize it as some linear distortion of the plane via scaling and rotation. I do not know how this distortion compares to the distortion that results from applying the transpose, or what one can say if the linear transformation is symmetric. Geometrically, why might we expect orthogonal matrices to be combinations of rotations and reflections?

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    the geometry is in the the inner product say on $\mathbb{R}^n$. the transpose satisfies $\langle Ax,y \rangle=\langle x,A^ty \rangle$. orthogonal matrices satisfy $\langle x,y \rangle=\langle Ax,Ay \rangle$, they preserve the geometry.2011-05-06
  • 0
    See [this answer](http://math.stackexchange.com/questions/598258/determinant-of-transpose/636198#636198) for a geometric description of the transpose.2014-01-12

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