Module 2: Highlights (abbreviated)

• Complex numbers:
  • Either $z = a + ib$ or $z = re^{i\theta}$
  • Conjugate: $z^* = a - ib = re^{-i\theta}$
  • Euler's: $e^{i\theta} = \cos\theta + i \sin\theta$
  • Rules for arithmetic.
  • From which, (complex-valued) functions $f(z)$

• Complex vectors:
  • Complex numbers as vector elements $$\kt{u} \eql \mat{1\\ 2-i\\ 3i} \;\;\;\;\;\; \kt{v} = \mat{2 \\ 1\\ i}$$
  • Conjugated row-vector: $$\eqb{ \br{u} & \eql & \kt{u}^\dagger & \eql & \mat{1 & 2+i & -3i} \\ \br{v} & \eql & \kt{v}^\dagger & \eql & \mat{2 & 1 & -i} }$$
  • Inner-product conjugates left side: $$\inr{u}{v} \eql \mat{1 & 2+i & -3i} \mat{2 \\ 1\\ i} \eql 7+i \;\;\;\; \mbx{A number}$$
  • Squared-magnitude (not length) of a complex vector: $\magsq{u} = \inr{u}{u}$
  • Outer-product: column times row

    

  • Scalar rules: $\kt{\alpha v} = \alpha\kt{v}$ and $\br{\alpha v} = \alpha^* \br{v}$
  • Inner products with linear combinations: $$\eqb{ \inrs{u}{\alpha v + \beta w} & \eql & \alpha\inr{u}{v} + \beta\inr{u}{w} & \mbx{Linearity on the right}\\ \inrs{\alpha u + \beta v}{w} & \eql & \alpha^*\inr{u}{w} + \beta^*\inr{v}{w} & \mbx{Conjugate linearity on the left}\\ }$$

Orthogonality, projectors:

• Vector orthogonality (defined as): $\inr{u}{v} = 0$

• Orthonormal:
  • $\inr{u}{v} = 0$
  • And $\mag{u} = \mag{v} = 1$

• A few important special 2D vectors: $$\eqb{ \kt{0} & \eql & \mat{1\\ 0} & \;\;\;\;\;\; & \kt{1} & \eql & \mat{0\\ 1}\\ \kt{+} & \eql & \mat{\isqt{1}\\ \isqt{1}} & \;\;\;\;\;\; & \kt{-} & \eql & \mat{\isqt{1}\\ -\isqt{1}} }$$

• Projections and projectors:
  • Let $\kt{v_1},\kt{v_2},\ldots$ be an orthonormal basis.
  • The projector for $\kt{v_1}$ (a matrix) is: $$P_{v_1} \eql \otr{v_1}{v_1}$$
  • The projection of any $\kt{u}$ on $\kt{v_1}$: $$\eqb{ P_{v_1} \kt{u} & \eql & \otr{v_1}{v_1} \kt{u} & \mbx{Apply projector} \\ & \eql & \kt{v_1} \; \inr{v_1}{u} & \mbx{Associativity} \\ & \eql & \inr{v_1}{u} \; \kt{v_1} & \mbx{Scalar movement} \\ }$$ The number $\inr{v_1}{u}$ is the coefficient of projection.
  • A vector is the sum of its projections: $$\kt{u} \eql \parenl{ \inr{v_1}{u} } \: \kt{v_1} + \ldots + \parenl{ \inr{v_n}{u} } \: \kt{v_n}$$
  • Projectors of a basis add up to the identity (completeness relation): $$\otr{v_1}{v_1} + \ldots + \otr{v_n}{v_n} \eql I$$

Operators:

• 3 types of operators: Hermitian, projectors, unitary
  • Hermitian: $A = A^\dagger$

    For example $$A \eql \mat{1 & -i\\ i & 1}$$ is Hermitian while $$B \eql \mat{i & 1\\ 1 & -i}$$ is not.

  • Unitary: when $A A^\dagger = A^\dagger A = I$

    For example $$H \eql \mat{\isqt{1} & \isqt{1}\\ \isqt{1} & -\isqt{1}}$$ is unitary because $H^\dagger H = H H^\dagger = I$
    But $$A \eql \mat{1 & -i\\ i & 1}$$ is not.

  • Projector: given a basis $\kt{v_1},\kt{v_2}\ldots\kt{v_n}$, a projector for $\kt{v_i}$ is the outer-product $$P_{v_1} \eql \otr{v_i}{v_i}$$

• Generalizing from real counterparts:
  • Hermitian generalizes real-symmetric
  • Projector works the same in complex/real
  • Unitary generalizes real-orthonormal

• "Dagger" properties:
  1. $\br{Ax} = \br{x} A^\dagger = \kt{A x}^\dagger$
  2. $\br{A^\dagger x} = \br{x} A$
  3. $\inr{w}{Ax} = \inr{A^\dagger w}{x}$
  4. $\inr{Aw}{x} = \inr{w}{A^\dagger x}$
  5. $(A^\dagger)^\dagger = A$
  6. $(\alpha A)^\dagger = \alpha^* A^\dagger$
  7. $(A + B)^\dagger = A^\dagger + B^\dagger$
  8. $(AB)^\dagger = B^\dagger A^\dagger$

• Hermitian properties: if $A,B$ are Hermitian
  1. $A + B$ is Hermitian.
  2. $\alpha A$ is Hermitian for real numbers $\alpha$.
  3. $A$'s diagonal elements are real numbers.
  4. $A$'s eigenvalues are real numbers.
  5. (Spectral theorem): one can find orthonormal eigenvectors that are a basis.
  6. $A$ can be written in terms of projectors made from the eigenvectors: $A = \sum_{i=1}^n \lambda_i \otr{v_i}{v_i}$.

• Properties of unitary operators: if $A, B$ are unitary
  1. $\inr{Au}{Av} = \inr{u}{v}$ (preserves inner products).
  2. $|Au| = |u|$ (preserves lengths).
  3. $A^\dagger$ and $A^{-1}$ are also unitary.
  4. The columns of $A$ are orthonormal, as are the rows.
  5. $AB$ and $BA$ are unitary.

• Operator sandwich:
  • $\swich{u}{A}{v} = \inrs{u}{Av} = \inrs{A^\dagger u}{v}$
  • Applied to a projector: $\swich{u}{P_v}{u} = \magsq{\inr{v}{u}} = \magsq{P_v\kt{u}}$

• Basis of the moment:
  • Vectors exist as mathematical objects without the numbers in them.
  • To put numbers to (i.e., "numerify") a vector, a basis must be selected.
  • It's much easier to select an orthonormal basis (any orthonormal basis).
  • We generally use the standard basis.
  • But some books will use the eigenbasis, which makes some calculations easier.

• Review of important and special 2D vectors: $$\eqb{ \kt{0} & \eql & \mat{1\\ 0} & & \\ \kt{1} & \eql & \mat{0\\ 1} & & \\ \kt{+} & \eql & \vectwo{ \isqt{1} }{ \isqt{1} } & \eql & \isqt{1} \mat{1\\ 0} \: + \: \isqt{1} \mat{0\\ 1} \\ \kt{-} & \eql & \vectwo{ \isqt{1} }{ -\isqt{1} } & \eql & \isqt{1} \mat{1\\ 0} \: - \: \isqt{1} \mat{0\\ 1} \\ }$$ Useful-to-know relationships between these: $$\eqb{ \kt{+} & \eql & \isqt{1} \kt{0} \: + \: \isqt{1} \kt{1} \\ \kt{-} & \eql & \isqt{1} \kt{0} \: - \: \isqt{1} \kt{1} \\ \kt{0} & \eql & \isqt{1} \kt{+} \: + \: \isqt{1} \kt{-} \\ \kt{1} & \eql & \isqt{1} \kt{+} \: - \: \isqt{1} \kt{-} \\ }$$