Matrices & linear systems
identity matrix
The identity matrix, written I, is the do-nothing matrix. It is square, with 1s running down the main diagonal and 0s everywhere else. For 2-by-2 it is [[1,0],[0,1]].
It plays the role that the number 1 plays for ordinary multiplication: just as 1 times any number leaves it unchanged, I times any matrix A gives back A, and A times I gives back A. As a transformation, I leaves every vector exactly where it was.
This makes I the natural starting point and reference for many ideas: a matrix and its inverse multiply to I, and asking 'does multiplying by this undo that?' always means 'do we get back to I?'
I = [[1,0],[0,1]], I * [[a,b],[c,d]] = [[a,b],[c,d]]
Multiplying by I changes nothing.
Only square matrices have an identity; an n-by-n matrix uses the n-by-n identity.
Also called
See also