What kind of transformation an upper triangular matrix represents

Every matrix represents a linear transformation, but depending on characteristics of the matrix, the linear transformation it represents can be limited to a specific type. For example, an orthogonal matrix represents a rotation (and possibly a reflection). Is it something similar about triangular matrices? Do they represent any specific type of transformation?
(Actually a reference describing different geometric transformations and their corresponding transformation matrices would be great)

• They preserve a chain of n subspaces stretching from 0 to the entire space, where n is the dimension. – rschwieb Jul 6 '14 at 12:05
• Actually I guess I should have said n+1 subspaces, to include 0! – rschwieb Jul 6 '14 at 12:47

Remember that the columns of a matrix are the images of a basis under the linear map that the matrix represents. The simplest observation for a triangular matrix is that the image of the $n$-h basis vector is in the span of the first $n$ basis vectors. So, the first vector gets mapped to somewhere on the line it generates, the second vector gets mapped into the plane generated by the first two vectors, and so on.

• +1 In other words, there's a chain of n subspaces such that the transformation maps each element of the chain into itself. This is the viewpoint that sticks with me best :) – rschwieb Jul 6 '14 at 12:01

For complex matrices, the Schur Theorem tells you that any matrix is unitarily equivalent to an upper triangular matrix. So, in a sense, all matrices are upper triangular.

Similarly, in the complex case, any nilpotent matrix can be represented by a strictly upper triangular matrix. So the strictly upper triangular matrices represent nilpotent matrices.

• Thanks, but I don't get your point. You mean there is a one-to-one mapping from upper triangular matrices and all other matrices?! But that sounds obviously wrong! – Alireza Mirian Jul 7 '14 at 10:08
• Any single matrix is unitarily equivalent to an upper triangular matrix, but the unitary transformation does not carry all matrices into upper triangular matrices! The theorem can, however, be extended to cover the case of a family of commuting matrices. – Jonas Dahlbæk Jul 7 '14 at 10:15

What you say about orthogonal matrices is true for dimensions 2 and 3. In higher dimensions an orthogonal matrix need neither be a reflection nor a rotation. (For example you can construct a $4\times4$ orthogonal matrix with two $2\times2$ orhtogonal matrices as diagonal blocks; both blocks could be reflections individually making the whole matrix of determinant 1; both could be rotations of different angles etc).

In dimension two a triangular matrix with 1's in the diagonal represents a shearing transformation (upper triangular means horizontal shear and lower triangular means vertical shear).

• I think in higher dimensions also orthogonal matrices represent rotation and reflection, but in higher dimension we just lose our geometric interpretation. However the concept is the generalization of the exact thing that we have in dimensions 2 and 3. – Alireza Mirian Jul 7 '14 at 10:05