Introduction: What Doesn't Change
The trace and rank are properties of a matrix that remain unchanged under certain transformations. The trace is the sum of diagonal elements; the rank is the number of independent rows (or columns). Both carry important physical meaning in quantum mechanics.
The Trace
The trace of a square matrix is the sum of its diagonal entries:
\[\text{Tr}(A) = \sum_{i=1}^{n} a_{ii}\]Key Properties
- Linearity: \(\text{Tr}(A + B) = \text{Tr}(A) + \text{Tr}(B)\)
- Cyclic property: \(\text{Tr}(ABC) = \text{Tr}(CAB) = \text{Tr}(BCA)\)
- Sum of eigenvalues: \(\text{Tr}(A) = \sum_i \lambda_i\)
- Basis independent: \(\text{Tr}(P^{-1}AP) = \text{Tr}(A)\)
The Rank
The rank of a matrix is the dimension of its column space (or row space):
- Maximum number of linearly independent columns
- Number of non-zero rows after row reduction
- Number of non-zero singular values
If rank(A) = n for an n×n matrix, then A is invertible.
Worked Examples
Example 1: Computing the Trace
\[A = \begin{pmatrix} 2 & 3 & 1 \\ 0 & -1 & 4 \\ 5 & 2 & 7 \end{pmatrix}\] \[\text{Tr}(A) = 2 + (-1) + 7 = 8\]Example 2: Cyclic Property
For 2×2 matrices \(A\) and \(B\):
\[\text{Tr}(AB) = \text{Tr}(BA)\]Even though \(AB \neq BA\), their traces are equal!
Example 3: Rank Determination
\[B = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 1 & 1 \end{pmatrix}\]Row 2 = 2 × Row 1, so rank(B) = 2 (not full rank, B is singular).
The Quantum Connection
The trace is central to quantum mechanics:
- Expectation values: \(\langle A \rangle = \text{Tr}(\rho A)\) for density matrix \(\rho\)
- Normalization: \(\text{Tr}(\rho) = 1\) for any valid quantum state
- Partial trace: Used to describe subsystems of entangled states
The cyclic property ensures expectation values don't depend on which basis you use.