Intuitive understanding of vector / matrix calculcation

Question

I am currently dealing with calculations done on vectors and matrices. For some parts I have gained an intuitive understanding, for others I didn't.

E.g., when we are adding two vectors, you can imagine that this means adding two arrows. The result is a single arrow that reflects the combined forces of the two individual source vectors. The resulting vector will probably have a new direction, which is influenced by the two original ones.

When we multiply a vector by a scalar, you can imagine that this means putting the very same arrow multiple times behind itself, to make it longer. I.e., the new vector has more force, but the direction stay the same.

Now… if I want to multiply a matrix by a vector, what is the analogy for that? What does this mean in terms of geometry?

Answer 1

Matrix multiplication generally is considered a sequence of simple geometric transformation:

1. Rotation
2. Scaling
3. effect of shearing, similar from elasticity.

The third effect would be better seen in terms of the coordinate tranformation. A matrix acts as a force to stress the cube formed by coordinate unit vectors.

Answer 2

Any matrix $A$ can be written as $A= U \Sigma V^T$ (the singular value decomposition), where $U,V$ are rotations, and $\Lambda$ has is a diagonal matrix with non negative entries.

While it may not bring any great enlightenment, you can consider the effect to be a rotation (which may not be a physically possible rotation) followed by scaling of each axis (possible to zero) followed by another rotation.

Answer 3

The matrix would represent a base, i.e. 2 vectors. With M = (B1,B2), doing VM is doing VxB1 + Vy*B2, i.e. plotting V coordinates in base B1,B2.

NB: For scalar multiplication, the intuition is more "scaling" that "making a copy", since the factor could be 3.7 or -1/pi :-) (or even complex).