let’s talk about matrices: linear operators, attitude matrices, and transition matrices. well, by “talk” i mean let’s play around, get our hands dirty.
ok, i’m in the xy-plane, i’ve got axes. i have a vector of length 2 at , and i want to rotate that vector by CCW (counter-clockwise). what’s the rotation matrix that does that?
here’s the vector.
now, i know i need ±sin 30 but i never remember where the negtive sign goes in the rotation matrix. in general, i know i need
(where i show that the sin terms have opposite signs, but which is where?) i also know that the columns of a matrix are the images of the basis vectors, so the first column
must be the image of (1,0) under the rotation; and that, i know, needs a positive y-component for a small CCW rotation. so the first column must have a positive sign. we write in general…
and, for in particular…
now we apply our rotation to the given vector…
so, we’re done. draw it.
OTOH, rotating a vector CCW should be the same as rotating the basis vectors CW. and that, i know how to write. that is, i know how to write the new basis vectors without hesitation. the new unit x-vector has components
and the new unit y-vector has components
(i wrote in the second one rather than because i’m working from an image in my head; analytically, i know that all 4 arguments should be the same angle.)
now i confirm that the new (red) basis is what i expect:
if i lay these two vectors out as rows, i get what is called the attitude matrix A for the new basis.
how do we use this to get the new components y of z, i.e. the components of z wrt the new basis? the relevant facts are
- (1) that the old components z are found by applying the transition matrix P to the new components y, and
- (2) the transition matrix is the transpose of the attitude matrix.
but we want y, so we write
so we need to compute , or we recognize that since – in this special case, A is a rotation, hence orthogonal – we have
, or .
we apply A to z:
is that the same? yes. what we had before was:
so we computed new components in two ways. rotating the vector is called an active transformation, or an “alibi” because the vector is elsewhere; rotating the basis out from under the vector is called a passive transformation, or an “alias” because it’s the same vector with different names (components).
when we change a basis, there’s only one matrix involved, although it too has an alias: if the new basis vectors are laid out in rows, it’s called an attitude matrix; if they’re laid out in columns, it’s called a transition matrix. one is the transpose of the other. to use one, you have to know which one it is.