Most matrices I have seen so far in real world (not that many actually) are either
– 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 i.e. 3-row 2-column.
– imagine a rectangle box
– 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