Day: March 1, 2021

Inner and Outer product in Linear Algebra

| Cosmoshivani

Let’s take two coordinate vectors, \textbf v and \textbf u. Both of them can be represented as column matrices (why not as row matrices?) of order m \times 1 and n \times 1.

The inner product of these two coordinate vectors is, \textbf v^T \cdot \textbf u.

\textbf v = \begin{pmatrix} a_1 \\ a_2 \\ a_3\\...\\a_m \end{pmatrix}

\textbf v^T = \begin{pmatrix} a_1 & a_2 & a_3 & ... & a_m \end{pmatrix}

\textbf u = \begin{pmatrix} b_1 \\ b_2 \\ b_3\\...\\ b_n \end{pmatrix}

\textbf v^T \cdot \textbf u = \begin{pmatrix} a_1 & a_2 & a_3& ... & a_m \end{pmatrix} \cdot \begin{pmatrix} b_1 \\ b_2 \\b_3 \\ ... \\b_n \end{pmatrix}

\textbf v^T \cdot \textbf u = ( a_1 b_1 + a_2 b_2 + a_3 b_3 + ... + a_m b_n)

Notice that it only works if m = n and it always results in a matrix of order 1 \times 1.

The outer product of these two vectors is \textbf v \cdot \textbf u^T.

\textbf v \cdot \textbf u^T = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \\ ... \\ a_m \end{pmatrix} \cdot \begin{pmatrix} b_1 & b_2 & b_3 & ... & b_n \end{pmatrix}

\textbf v \cdot \textbf u^T = \begin{pmatrix} a_1 b_1 & a_1 b_2 & a_2 b_3 &...&a_1 b_n\\a_2 b_1 & a_2 b_2 & a_2 b_3 &...& a_2 b_n\\a_3 b_1 & a_3 b_2 & a_3 b_3 &...&a_3 b_n\\...& ... & ... & ... & ... \\a_m b_1 & a_m b_2 & a_m b_3 & ... & a_m b_n \end{pmatrix}

If you have a column vector of order m \times 1 and a row vector of order 1 \times n respectively, their outer product will be a matrix of order m \times n. Here m need not be equal to n.

The significance of inner product in quantum mechanics becomes clear right away, but what about the outer product?

To be continued in the next article: Inner and Outer Product in Quantum Mechanics