Confusion regarding taking the Hermitian adjoint of an outer product in spectral decomposition

From my notes$${}^{\zeta}$$, I have that, So in short, I don't understand why, when we take the Hermitian conjugate of an outer product, say, $$\lvert A_i \rangle\langle A_i \rvert$$ it seems like we should just end up with an inner product, say, $$\langle A_i \lvert A_i \rangle$$ since I am just applying the result of eqn $$(3.30)$$ above, which sends $$\lvert A_i \rangle$$ to $$\langle A_i \rvert$$, and vice versa, which is tantamount to switching the order of them.

So, a more careful analysis is required... I don't understand how eqn $$(4.15)$$ follows from eqn $$(4.14)$$, specifically, I know that for Hermitian operators $$\hat{A}^{\dagger}=\hat{A}^*=\hat A$$, since the eigenvalues of Hermitian operators are real. I understand this because the Hermitian adjoint, $$\dagger$$ operation means to transpose first, $$\intercal$$, then take the complex conjugate. But since there is only one operator, $$\hat{A}$$, we must have the case that $$\hat{A}^{\intercal}=\hat{A}$$.

In an attempt to understand this I will use 2 different indices, $$i,j$$, (as the fact that the same index was being used in my notes may be the source of my confusion). Starting with $$(4.14)$$, but only focussing on the outer product part, $$\lvert A_i \rangle\langle A_j \rvert$$ So if I can demonstrate that $$\lvert A_i \rangle\langle A_j \rvert$$ is invariant under $$\dagger$$, then I will have understood it.

I start by taking the Hermitian conjugate $$\Bigl(\lvert A_i \rangle\langle A_j \rvert\Bigl)^{\dagger}\stackrel{\intercal}{=}\Bigl(\lvert A_i \rangle\langle A_j \rvert\Bigl)^{\intercal}=\langle A_j \lvert A_i \rangle\stackrel{*}=\color{blue}{\Bigl(\langle A_j \lvert A_i \rangle\Bigl)^{*}=\langle A_i \lvert A_j \rangle}\ne \lvert A_i \rangle\langle A_j \rvert$$

In the blue part above I am making use of eqn $$(3.27)$$, which essentially states that, $$\Bigl(\langle \beta \mid \gamma\rangle\Bigl)^*=\langle \gamma \mid \beta\rangle$$ for Hilbert space vectors $$\beta, \gamma$$.

So in short, I am asking why, under the Hermitian adjoint operation, eqn $$(4.14)$$, $$\hat {A}=\sum_i A_i\lvert A_i\rangle\langle A_i \rvert$$ doesn't become $$\sum_i {A_i}^* \langle A_i \mid A_i \rangle?$$

$${}^{\zeta}$$ Sources from ICL dept. of physics

• You should use MathJax to type the most important equations. It seems that you only need two or three equations, so please consider to type them instead of uploading (very large) images. Jun 18 '21 at 12:04
• @Jakob "You should consider to use MathJax to write the most important equations", I already considered that and explicitly typed the ones I thought most important. If others are more important then please let me know and I will type those out also, thanks. Jun 18 '21 at 12:06
• You uploaded two large images with roughly ten equations, of which only two or three are important. Jun 18 '21 at 12:07
• @Jakob What did I just write above? Jun 18 '21 at 12:08
• @Jakob I got no problem doing what you say, if you tell me the eqn no.s to be typed out I will do it. Jun 18 '21 at 12:13

Let's first look at what happens if we take the Hermitian conjugate of a ket: $$|A_i\rangle^\dagger = \langle A_i|.$$ If we have two operators $$C$$ and $$D$$, and want to take the Hermitian conjugate of their product, we have to reverse their order and take the Hermitian conjugates of $$C$$ and $$D$$: $$(CD)^\dagger = D^\dagger C^\dagger$$ Now to your question: If we have a ket-bra and want to take the Hermitian conjugate, we again have to reverse the order, and then take the Hermitian conjugate of the ket and bra individually: $$(|A_i\rangle \langle A_j|)^\dagger = (\langle A_j|)^\dagger (|A_i\rangle)^\dagger = |A_j\rangle\langle A_i|$$ Does this answer your question?

• Yes, this answers my question, sorry for the slow reply, been busy. The key word here is that Hermitian conjugation must be applied individually, what I was doing wrong in my question was applying it to the combined object. Many thanks. Jul 2 '21 at 16:54

The projector $$P_{A_i}=|A_i\rangle\langle A_i|$$ is an operator, and the scalar product $$\langle A_i|A_i\rangle$$ is a complex number. If you do the adjoint of an operator you should get another operator, not a number. Thinking of them in their matrix form, transposing a matrix and then complex conjugating each entry gives another matrix, not a number. Sure, you can apply $$|A_i\rangle^\dagger=\langle A_i|$$ and $$\langle A_i|^\dagger=|A_i\rangle$$, but you have to be careful with the order, so that after doing the adjoint you still get an operator.

Another way of seeing that in general $$(|\alpha\rangle\langle\beta|)^\dagger=|\beta\rangle\langle\alpha|$$ is this: by definition, $$\langle\phi|O^\dagger|\psi\rangle$$ is the scalar product of $$O|\phi\rangle$$ and $$|\psi\rangle$$. Therefore it is also the scalar product of $$|\psi\rangle$$ and $$O|\phi\rangle$$ complex conjugated

$$\langle\phi|O^\dagger|\psi\rangle=\langle\psi|O|\phi\rangle^*.\tag{1}$$

So in particular, for an operator of the form $$O=|\alpha\rangle\langle\beta|$$ we have

\begin{align} \langle\psi|O|\phi\rangle^*&=\left(\langle\psi|\alpha\rangle\langle\beta|\phi\rangle\right)^*\\ &=\langle\psi|\alpha\rangle^*\langle\beta|\phi\rangle^*\\ &=\langle\alpha|\psi\rangle\langle\phi|\beta\rangle\\ &=\langle\phi\underbrace{|\beta\rangle\langle\alpha|}_O^\dagger}\psi\rangle\tag{2},\\ \end{align so by comparison with $$(1)$$, $$(|\alpha\rangle\langle\beta|)^\dagger=O^\dagger=|\beta\rangle\langle\alpha|$$.

Notations:

\begin{align} A_i &: \text{ is a real number}.\\ \big\vert A_i \rangle &: \text{ a column vector}.\\ \langle A_i \big\vert =\big\vert A_i \rangle^\dagger &: \text{ a row vector and complex adjoint}. \\ \big\vert A_i \rangle \langle A_i \big\vert &: \text{ a projection matrix}.\\ \langle A_i \big\vert A_i \rangle &= \text{ a real number}. \end{align}

And note the transporse of product of two matrix (or vector, a special matrix): $$\left(\mathbf{A} \mathbf{B}\right)^T = \mathbf{B}^T \mathbf{A}^T$$ After transporse, the order of product is reversed.

Therefore, for the matrix $$\mathbf{M}$$:

$$\mathbf{M} = \sum_i \lambda_i \big\vert U \rangle \langle V \big\vert$$

It is transport:

\begin{align} \mathbf{M}^\dagger &= \left\{\sum_i \lambda_i^* \big\vert U \rangle \langle V \big\vert \right\}^\dagger\\ &= \sum_i \lambda_i^* \left(\langle V \big\vert\right)^\dagger \left(\big\vert U \rangle \right)^\dagger\\ &= \sum_i \lambda_i^* \big\vert V \rangle \langle U \big\vert \end{align}

• Thanks for your answer, does "transport"=transpose? Jun 18 '21 at 12:46
• Yes. The post corrected.
– ytlu
Jun 18 '21 at 13:33