5 ms·
The first three can reasonably be thought of as defining linear transformations. For linear systems of equations A x = b, x is an unknown vector in the input sp
by obastani 7y ago
The first three can reasonably be thought of as defining linear transformations. For linear systems of equations A x = b, x is an unknown vector in the input space that is mapped by A to b.
Both covariance matrices and Hessians are more naturally thought of as tensors, not matrices (and therefore not linear transformations). That is, they take in two vectors as input and produce a single real number as output.
As for graph adjacency matrix, this can actually be thought of as a linear transformation on the vector space where the basis vectors correspond to nodes in the graph. Linear combinations of these basis vectors correspond to probability distributions over the graph (if properly normalized).
2D images... Yes, these cannot really be interpreted as linear transformations. But I'd say these aren't really matrices in the mathematical sense.
- xscott 7y agoIf you squint hard enough, you can see all of them as linear transformations (even the 2D images :-). I politely disagree about covariance and Hessians. I can squint and say that the Hessian provides a change in gradient when multiplied by a delta vector. Similarly for covariance... Or you could look at it as one half of the dot product for a Bhattacharyya distance, which is just a product of three matrices (row vector, square matrix, col vector). No need for tensors yet. That is unless you decide to squint hard enough to see everything as tensors! :-)