# Linear Algebra

**9.3 Rotation matrices**

Here's our old example of rotating coordinate axes.
= (x, y) is a vector.
Let's

call the vector in the new coordinate system. The two are
related by

Figure 1: Rotation of coordinate axes by θ.

the equations of coordinate transformation we discussed in week 4 of the course.

These may be written in matrix form in a very convenient way (check):

where is a rotation matrix by
θ. Note the
transformation preserves lengths

of vectors as we mentioned before. This means the rotation matrix
is

orthogonal:

These matrices have a special property ("group property"),
which we can show

by doing a second rotation by θ':

Thus the transformation is linear. More general def. .

**9.4 Matrices acting on vectors**

or more compactly

Another notation you will often see comes from the early
days of quantum mechan-

ics. Write the same equation

So this is a "ket", or column vector. A "bra", or row
vector is written as the adjoint

of the column vector:

N.B. In this notation the scalar product of
and is
, and the length

of a vector is given by .

**9.5 Similarity transformations
**

Suppose a matrix B rotates to . Now we rotate the coordinate

system by some angle as well, so that the vectors in the new system are and

e.g. What is the matrix which relates to i.e. the transformed

matrix B in the new coordinate system?

so the matrix B in the new basis is

This is called a similarity transformation of the matrix
B.

To retain:

• similarity transformations preserve traces and determinants:,

detM = detM'.

• matrices R which preserve lengths of real vectors are called orthogonal, RR^{T} =

1 as we saw explicitly.

• matrices which preserve lengths of complex vectors are called unitary. Suppose

, require , then

A similarity transformation with unitary matrices is
called a unitary transfor-

mation.

• If a matrix is equal to its adjoint, it's called self-adjoint or Hermitian.

Examples

1.

is symmetric, i.e. A = A^{T} , also Hermitian because it is
real.

2.

is antisymmetric, and anti-self-adjoint, since .

3.

is Hermitian, .

4.

is antiHermitian, . Check!

5.

is unitary, .

6.

is orthogonal, A^{-1} = A^{T} . Check!

**9.5.1 Functions of matrices
**

Just as you can define a function of a vector (like ), you can define a function

of a matrix M, e.g. F(M) = aM

^{2}+ bM

^{5}where a and b are constants. The

interpretation here is easy, since powers of matrices can be understood as repeated

matrix multiplication. On the other hand, what is exp(M)? It must be understood

in terms of its Taylor expansion, !. Note that this makes no

sense unless every matrix element sum converges.

Remark: note unless [A,B] = 0! Why? (Hint: expand both sides.)