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- Thread starter monea83
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chiro

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In general there is no real interpretation that comes to mind. With some specific groups the transpose matrix is actually the inverse matrix. Matrices that have this property include the rotation matrices that have a determinant of one.

Geometrically rotation matrices conserve length. So if you had a vector with the tail at the origin (in other words a point), then when you apply a rotation it preserves length of that vector: if you apply it to a set of points it preserves area/volume etc.

There are of course other uses for transpose matrices like least squares, but off the top of my head I can't give you geometric descriptions or interpretations for those.

- #3

Hurkyl

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Lots of practice.is there a way to understand them intuitively,

One thing I intuit about them is that it is often the "right" thing to use when you might have used x

Also, you can think of them as being a way to turn a matrix into a symmetric, square matrix that does the "least damage" in some sense.

Of course, these are all algebraic ways to intuit them, rather than the geometric one you asked about.

- #4

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For n=3 for exemple,

Think at A as a linear transformation on R3: then you can think at A^T A as a convex bilinear transformation on R3xR3 giving you the scalar product of two vectors A(x1) and A(x2)

In particular, when x1 = x2 = x, I think at it as a "lengh" fonction on A(x) (hence positive defiinte)

- #5

Landau

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The Singular value decomposition has some geometric implications, but I don't know whether this qualifies as a geometric explanation of A*A itself.

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