– square matrices or

– column/row vectors

However it is good to develop a really quick and intuitive feel for matrix shape. When you are told there’s some mystical 3×2 matrix,

– on its left imagine a vertical keyboard

– on it put Left fingers, curling. 3 fingers only.

– Next, imagine a horizontal keyboard below (or above, if comfortable) the rectangle.

– put Right fingers there. 2 fingers only

For me, this gives a physical feel for the matrix size. Now let’s try it on a column matrix of 11. The LHS vertical keyboard is long – 11 fingers. Bottom keyboard is very short — 1 finger only. So it’s 11×1

The goal is to connect the 3×2 (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

————————

Now consider matrix multiplication. Consider a 11×2. Note a 11×1 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 11×1 columnar matrix X, which is more common.

X * X’ is like what we just showed — 11×11 matrix matrix

X’ * X is 1×11 multiplying 11×1 — 1×1 tiny matrix