Most matrices I have seen so far in real world (not that many actually) are either

- square matrices or

- column/row vectors- square matrices or

- imagine a rectangle box

- on its left imagine a vertical keyboard- on it put Left fingers, curling. 3 fingers only.

The goal is to connect the 3x2 (abstract) notation to the visual layout. To achieve that,

- I connect the notation to -- the hand gesture, then to -- the visual. Conversely,

- I connect the visual to -- the hand gesture, then to -- the notation

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Now consider matrix multiplication. Consider a 11x2. Note a 11x1 columnar matrix is more common, but it's harmless to get a more general feel.

An 11x2 * 2x9 gives a 11x9.

Finger-wise, the left 11 fingers on the LHS matrix stay glued; and the right 9 fingers on the RHS matrix stay glued. In other words,

The left hand fingers on the LHS matrix remain.

The right hand fingers on the RHS matrix remain.

Consider a 11x1 columnar matrix X, which is more common.

X * X' is like what we just showed -- 11x11 matrix matrix

X' * X is 1x11 multiplying 11x1 -- 1x1 tiny matrix

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