# Linear Algebra: #19 "Classical Groups" often seen in Physics

- The orthogonal group
*O*(n): This is the set of all linear mappings f : ℜ^{n}→ ℜ^{n}such that <**u**,**v**> = <f(**u**), f(**v**)>, for all**u, v**∈ ℜ^{n}. We think of this as being all possible rotations and inversions (Spiegelungen) of n-dimensional Euclidean space. - The special orthogonal group
*SO*(n): This is the subgroup of*O*(n), containing all orthogonal mappings whose matrices have determinant +1. - The unitary group
*U*(n): The analog of*O*(n), where the vector space is n-dimensional complex space ℂ^{n}. That is, <**u**,**v**> = <f(**u**), f(**v**)>, for all**u, v**∈ ℂ^{n}. - The special unitary group
*SU*(n): Again, the subgroup of*U*(n) with determinant +1.

Note that for orthogonal, or unitary mappings, all eigenvalues — if they exist — must have absolute value 1. To see this, let

**v**be an eigenvector with eigenvalue λ. Then we have

Since

**v**is an eigenvector, and thus

**v**≠

**0**, we must have |λ| = 1.

We will prove that all

*unitary*matrices can be diagonalized. That is, for every unitary mapping ℂ

^{n}→ ℂ

^{n}, there exists a basis consisting of eigenvectors. On the other hand, as we have already seen in the case of simple rotations of 2-dimensional space, “most” orthogonal matrices cannot be diagonalized. On the other hand, we can prove that every orthogonal mapping ℜ

^{n}→ ℜ

^{n}, where n is an

*odd*number, has at least one eigenvector. [For example, in our normal 3-dimensional space of physical reality, any rotating object — for example the Earth rotating in space — has an axis of rotation, which is an eigenvector.]

- The self-adjoint mappings f (of ℜ
^{n}→ ℜ^{n}or ℂ^{n}→ ℂ^{n}) are such that <**u**, f(**v**)> = <f(**v**),**u**>, for all**u, v**in ℜ^{n}or ℂ^{n}, respectively. As we will see, the matrices for such mappings are symmetric in the real case, and*Hermitian*in the complex case. In either case, the matrices can be diagonalized. Examples of Hermitian matrices are the Pauli spin-matrices:

We also have the

*Lorentz group*, which is important in the Theory of Relativity. Let us imagine that physical space is ℜ

^{4}, and a typical point is

**v**= (t

_{v}, x

_{v}, y

_{v}, z

_{v}). Physicists call this

*Minkowski space*, which they often denote by M

^{4}. A linear mapping f : M

^{4}→ M

^{4}is called a

*Lorentz transformation*if, for f(

**v**) = (t

_{v}

^{*}, x

_{v}

^{*}, y

_{v}

^{*}, z

_{v}

^{*}), we have

- − (t
_{v}^{*})^{2}+ (x_{v}^{*})^{2}+ (y_{v}^{*})^{2}+ (z_{v}^{*})^{2}= − t_{v}^{2}+ x_{v}^{2}+ y_{v}^{2}+ z_{v}^{2}for all**v**∈ M^{4}, and also the mapping is “time-preserving” in the sense that the unit vector in the time direction, (1, 0, 0, 0) is mapped to some vector (t^{*}, x^{*}, y^{*}, z^{*}), such that t^{*}> 0.

The

*Poincare group*is obtained if we consider, in addition, translations of Minkowski space. But translations are not linear mappings, so I will not consider these things further in this lecture.

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