Notation « Brainrack Aborning – Developing a Search Engine

Bra-Ket Matrix Notation

February 11, 2008

We’ll need a notation for matrices and vectors. The standard notation for a column vector is $\mathbf{a}=\underline{a}=\left\{a_1,a_2,\ldots,a_n\right\}^T$
whilst a row vector is written using transpose notation $\underline{a}^T$ .

The standard notation for a matrix and a matrix composed of vectors is
$\mathbf{A}=\underline{\underline{A}}=\begin{pmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{pmatrix} = \begin{bmatrix} \underline{a}_1^T \\ \vdots \\ \underline{a}_m^T \end{bmatrix} = \begin{bmatrix} \underline{a}_1 \cdots \underline{a}_n \end{bmatrix}.$

Unfortunately, this notation will be a bit awkward for my purposes. I will instead abuse Dirac’s bra-ket notation to write vectors and matrices in summation form as follows:

$\mathbf{a}=\sum_{1 \le i \le n}\left\rvert \hat{\imath} \right\rangle a_i$

$\mathbf{a}^T=\sum_{1 \le j \le m} a_j \left\langle \hat{\jmath} \right\lvert$

$\mathbf{A}=\sum_{\substack{1 \le i \le n \\ 1 \le j \le m}} \left\rvert \hat{\imath} \right\rangle a_{ij} \left\langle \hat{\jmath} \right\lvert.$

The rule for projecting unit vectors is:

$\left\langle \hat{\imath} \right\lvert \left\rvert \hat{\jmath} \right\rangle =\delta_{ij} =\left\{ \begin{array}{ll} 0 & i \ne j \\ 1 & i = j \end{array} \right.$

Bra $\left\langle \right\lvert$ and ket $\left\rvert \right\rangle$ do not commute with each other; though they do commute with numbers.

To illustrate, here is the multiplication of two matrices in this notation

$\mathbf{A}=\sum_{\substack{1 \le i \le n \\ 1 \le j \le m}} \left\rvert \hat{\imath} \right\rangle a_{ij} \left\langle \hat{\jmath} \right\lvert \text{;} \mathbf{B}=\sum_{\substack{1 \le u \le m \\ 1 \le v \le p}} \left\rvert \hat{u} \right\rangle b_{uv} \left\langle \hat{v} \right\lvert$

$\mathbf{A}\times\mathbf{B} = \sum_{i,j} \left\rvert \hat{\imath} \right\rangle a_{ij} \left\langle \hat{\jmath} \right\lvert \times \sum_{u,v} \left\rvert \hat{u} \right\rangle b_{uv} \left\langle \hat{v} \right\lvert$

$= \sum_{i,j,u,v} \left\rvert \hat{\imath} \right\rangle a_{ij} \left\langle \hat{\jmath} \right\lvert \left\rvert \hat{u} \right\rangle b_{uv} \left\langle \hat{v} \right\lvert$

$= \sum_{i,j,u,v} \left\rvert \hat{\imath} \right\rangle a_{ij} \delta_{ju} b_{uv} \left\langle \hat{v} \right\lvert$

$= \sum_{i,j,v} \left\rvert \hat{\imath} \right\rangle a_{ij} b_{jv} \left\langle \hat{v} \right\lvert$

$= \sum_{i,v} \left\rvert \hat{\imath} \right\rangle \left\langle \hat{v} \right\lvert \sum_{j} a_{ij} b_{jv}$

Here is the usual inner product between two vectors

$\mathbf{a} \circ \mathbf{b} = \mathbf{a}^T \mathbf{b} = \sum_{i} a_i \left\langle \hat{\imath} \right\lvert \sum_{j} \left\rvert \hat{\jmath} \right\rangle b_j = \sum_{i,j} a_m \delta_{i,j} b_j = \sum_{i} a_i b_i$