# Defining an inner product over matrices and over vectors

In quantum mechanics, in Dirac notation an inner product is denoted as $$\langle A|B\rangle$$ and one fundamental postulate is given as follows:

$$\langle A|B\rangle = \langle B|A\rangle ^*$$

If I were to consider $$|A\rangle$$ and $$|B\rangle$$ as vectors represented by column matrices, then the inner product between two vectors (column matrices) is defined as $$A^TB$$. If the vectors are complex then, the inner product is defined to be $$A^{*T}B$$ where the $$*$$ represents complex conjugation.

I am unable to put these two definitions together, due to the following confusion.

For any vector $$|A\rangle$$ , I take $$\langle A|$$ to be the conjugate transpose, which would be given as $$A^{*T}$$.

Therefore, if I convert the postulate in dirac notation to matrix algebra, I would have the LHS to be $$A^{*T}B$$. Taking a transpose, I would have:

$$(A^{*T}B)^T = B^TA^*$$

Now taking, the complex conjugate, I would get:

$$(A^{*T}B)^{*T} = B^{*T}A$$

And hence, $$((A^{*T}B)^{*T})^{*T} = A^{*T}B= (B^{*T}A)^{*T}$$

But according to the earlier postulate, I would have:

$$A^{*T}B = (B^{*T}A)^*$$

It seems like I am missing a transpose. I suspect I am going wrong with my analogy to matrix representation, but I do not understand where. I'm terribly sorry about my rudimentary LaTex and formatting skills.

• Some of your transposes are on scalars. Sep 15 '20 at 17:21

The result of an inner product is a scalar--that is, a 1x1 matrix. Transposing a scalar leaves it unchanged. So, $$(B^{*T}A)^{*T} = (B^{*T}A)^{*}$$.

On another note, the usual symbol for the complex transpose is the dagger $$^{\dagger}$$, \dagger in LaTeX.