Videos for Section 7.1


In $\mathbb{C}^n$, the inner product $\langle {\bf x}| A {\bf y} \rangle$ is a product of three terms: $$\langle {\bf x}| A {\bf y} \rangle = \begin{matrix} (\bar x_1, \bar x_2, \ldots, \bar x_n ) \cr \phantom{asssdf} \cr \phantom{asdf} \end{matrix} \begin{pmatrix} \phantom{a} & & \phantom{a} \cr & A & \cr \phantom{a} & & \phantom{a} \end{pmatrix} \begin{pmatrix} y_1 \cr \vdots \cr y_n \end{pmatrix} $$ The notation $\langle {\bf x} | A {\bf y} \rangle$ implies that we first multiply $A$ times ${\bf y}$, then then multiply $\bar {\bf x}^T$ times $A {\bf y}$. But we could just as well first multiply ${\bf x}^T$ by $A$, then take the product of that with ${\bf y}$. In other words, when it comes to inner products, there is an operation on ${\bf x}$ that is equivalent to applying $A$ to ${\bf y}$.

In general, if $V$ is a vector space with an inner product and $L: V \to V$ is a linear operator, then the adjoint of $L$, denoted $L^\dagger$ (pronounced $L$-dagger), is an operator such that $$ \langle L^\dagger {\bf x} | {\bf y} \rangle = \langle {\bf x} | L {\bf y} \rangle$$ for every pair of vectors ${\bf x}, {\bf y} \in V$. Some general properties of adjoints include:

  1. $L^\dagger$ always exists.
  2. $(L+M)^\dagger = L^\dagger + M^\dagger$
  3. $(cL)^\dagger = \bar c L^\dagger$ (not $c L^\dagger$!)
  4. $(LM)^\dagger = M^\dagger L^\dagger$
  5. $(L^\dagger)^\dagger = L$
For $\mathbb{R}^n$ or $\mathbb{C}^n$ with the standard inner product, operators are just square matrices, and adjoints are transpose conjugates: $$A^\dagger = \bar A^T$$ These properties and formulas are developed in the first video:

In the second video, we consider adjoints on general vector spaces. If $V$ is a vector space with an orthonormal basis $\mathcal{B}$, then $$[L^\dagger]_{\mathcal{B}} = \overline{[L]_{\mathcal{B}}^T}.$$ From this we get an explicit formula for $L^\dagger$. On the vector space $L^2(\mathbb{R})$, integration by parts implies that the adjoint of $d/dt$ is $-d/dt$. In quantum mechanics we often study the momentum operator $\hat p = -i\hbar d/dx$, whose adjoint is itself.