3.1. Matrices and vectors

Matrices and vectors are convenient tools for representing linear systems with compact notation that can be processed by a computer.

Example: Flash distillation

You are analyzing the molar flow rates resulting from a flash distillation process.

The steady-state mole balances for the total process and component A are

(3.1)\[\begin{align} 10 &= \dot L + \dot V \\ 5 &= 0.3 \dot L + 0.8 \dot V \end{align}\]

These equations can equivalently be represented using matrices and vectors as

(3.2)\[\begin{equation} \begin{bmatrix} 10 \\ 5 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 0.3 & 0.8 \end{bmatrix} \begin{bmatrix} \dot L \\ \dot V \end{bmatrix} \end{equation}\]

We will learn why these representations are equivalent shortly. Importantly, because our mole balances can be written in this way, they are a system of linear equations that can be solved using techniques of linear algebra.

3.1.1. Definition

A matrix is a rectangular array of quantities, which we call its elements, that are laid out in horizontal rows and vertical columns. We will typically denote a matrix by a bold, capital letter such as A.

(3.3)\[\begin{equation} \vv{A} = \begin{bmatrix} 0.3 & 1 & -5 \\ 0 & -0.2 & 16 \end{bmatrix} \end{equation}\]

An m x n matrix has m rows and n columns. A is a 2 x 3 matrix. We will sometimes refer to elements of a matrix by their row and column

(3.4)\[\begin{equation} \vv{A} = \begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \end{bmatrix} \end{equation}\]

For example, \(A_{12} = 1\) and \(A_{23} = 16\) for A given above.

A vector is a matrix with either one column (a column vector) or one row (a row vector). We will typically denote a vector by a bold, lowercase letter such as b

(3.5)\[\begin{equation} \vv{b} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \qquad \vv{c} = \begin{bmatrix} -1 & 0 & 1 \end{bmatrix} \end{equation}\]

b is a 2-element column vector that is also a 2 x 1 matrix, while c is a 3-element row vector that is also a 1 x 3 matrix. When referring to elements of a vector, it typical to only use one index

(3.6)\[\begin{equation} \vv{b} = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix} \qquad \vv{c} = \begin{bmatrix} c_1 & c_2 & c_3 \end{bmatrix} \end{equation}\]

For b and c given above, \(b_2 = 2\) and \(c_2 = 0\). Note that it becomes ambiguous whether you have a column vector or a row vector if you are refering to elements in this way.

There are some other types of “special” matrices.

• Square matrix: a matrix with equal numbers of rows and columns (an n x n matrix).

  (3.7)\[\begin{equation} \begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} \end{equation}\]

• Diagonal matrix: a square matrix with nonzero entries only for the elements on the diagonal, \(A_{ii}\) for \(i = 1, ..., n\).

  (3.8)\[\begin{equation} \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} \end{equation}\]

• Upper triangular matrix: a square matrix with nonzero entries only on the diagonal or above, \(A_{ij}\) for \(i = 1, ..., n\) and \(j \ge i\). on the diagonal, \(A_{ii}\).

  (3.9)\[\begin{equation} \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix} \end{equation}\]

• Lower triangular matrix: a square matrix with nonzero entries only on the diagonal or below, \(A_{ij}\) for \(i = 1, ..., n\) and \(j \le i\).

  (3.10)\[\begin{equation} \begin{bmatrix} 1 & 0 \\ 4 & 3 \end{bmatrix} \end{equation}\]

• Identity matrix: a diagonal matrix of ones, typically denoted I.

  (3.11)\[\begin{equation} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \end{equation}\]

There are several other types of special matrices, but we will leave those for a longer course on linear algebra!