Matrices and vectors

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.

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!

3.1.2. Addition and scalar multiplication#

  • Equality: \(\vv{A} = \vv{B}\) if and only if A and B have the same size and all their corresponding elements are equal.

  • Addition: \(\vv{C} = \vv{A} + \vv{B}\) is defined if A and B have the samesize. Then, the matrix elements are added element-wise:

    (3.12)#\[\begin{equation} C_{ij} = A_{ij} + B_{ij} \end{equation}\]

    Example:

    (3.13)#\[\begin{equation} \begin{bmatrix} 5 & -1 & 0 \\ 3 & 1 & 0 \end{bmatrix} + \begin{bmatrix} 1 & 5 & 3 \\ 3 & 2 & 2 \end{bmatrix} = \begin{bmatrix} -4 & 6 & 3 \\ 0 & 1 & 2 \end{bmatrix} \end{equation}\]

  • Scalar multiplication: \(\vv{B} = k\vv{A}\) multiplies each element of A by k:

    (3.14)#\[\begin{equation} B_{ij} = kA_{ij} \end{equation}\]

    Example:

    (3.15)#\[\begin{equation} -2 \begin{bmatrix} 1 & 0 \\ 0 & -2 \end{bmatrix} = \begin{bmatrix} -2 \times 1 & -2 \times 0 \\ -2 \times 0 & -2 \times 4 \end{bmatrix} = \begin{bmatrix} -2 & 0 \\ 0 & 4 \end{bmatrix} \end{equation}\]

  • Subtraction: same as addition of negative \(\vv{C} = \vv{A} - \vv{B} = \vv{A} + (-\vv{B})\) so

    (3.16)#\[\begin{equation} C_{ij} = A_{ij} - B_{ij} \end{equation}\]

Example: Matrix addition and scalar multiplication

Given

(3.17)#\[\begin{equation} \vv{A} = \begin{bmatrix} -1 & 2 \\ 0 & 5 \end{bmatrix} \qquad \vv{B} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \end{equation}\]

Compute \(2\vv{A} - \vv{B}\).

(3.18)#\[\begin{align} 2\vv{A}-\vv{B} &= 2 \begin{bmatrix} -1 & 2 \\ 0 & 5 \end{bmatrix} - \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \\ &= \begin{bmatrix} -2 & 4 \\ 0 & 10 \end{bmatrix} - \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \\ &= \begin{bmatrix} -3 & 4 \\ 0 & 9 \end{bmatrix} \end{align}\]

3.1.3. Matrix multiplication#

C = AB is defined if A has the same number of columns as B has rows. If A is m x p and B is p x n, C is m x n and its elements are

(3.19)#\[\begin{equation} C_{ij} = \sum_{k=1}^p A_{ik} B_{kj} \end{equation}\]

Example:

(3.20)#\[\begin{align} \begin{bmatrix} 3 & 5 \\ 4 & 0 \\ -6 & -3 \end{bmatrix} \begin{bmatrix} 2 & -2 \\ 5 & 0 \end{bmatrix} &= \begin{bmatrix} 3\times2 + 5\times5 & 3\times-2 + 5\times0 \\ 4\times2 + 0\times5 & 4\times-2 + 0\times0 \\ -6\times2 + -3\times5 & -6\times-2 + -3\times0 \end{bmatrix} \\ &= \begin{bmatrix} 31 & -6 \\ 8 & -8 \\ -27 & 12 \end{bmatrix} \end{align}\]

Multiplying with a vector works the same!

(3.21)#\[\begin{align} \begin{bmatrix} 3 & 5 \\ 4 & 0 \\ -6 & -3 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \end{bmatrix} &= \begin{bmatrix} 3 \times 1 + 5 \times 2 \\ 4 \times 1 + 0 \times 2 \\ -6 \times 1 + -3 \times 2 \end{bmatrix} \\ &= \begin{bmatrix} 8 \\ 4 \\ -12 \end{bmatrix} \end{align}\]

3.1.4. Transpose#

\(\vv{A}^{\rm T}\) is the transpose of A, and its elements are obtained by “flipping” the rows and columns:

(3.22)#\[\begin{equation} A_{ij}^{\rm T} = A_{ji} \end{equation}\]

Example:

(3.23)#\[\begin{equation} \begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix}^{\rm T} = \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix} \end{equation}\]

A matrix is called symmetric if \(\vv{A}^{\rm T} = \vv{A}\).

3.1.5. Skill builder problems#

Given the matrices:

\[\begin{split} \vv{A} = \begin{bmatrix} 0 & 2 \\ 2 & 4 \\ 1 & 3 \\ \end{bmatrix} \quad \vv{B} = \begin{bmatrix} 0 & 2 &1 \\ 2 & 4 & 3 \end{bmatrix} \quad \vv{C} = \begin{bmatrix} 3 & 0 & 4 \\ -1 & 2 & 2 \\ 6 & 5 & -4 \end{bmatrix} \quad \vv{D} = \begin{bmatrix} 0 & -5 & -3 \\ -5 & 2 & 4 \\ -3 & 4 & 0 \end{bmatrix} \end{split}\]

and vectors:

\[\begin{split} \vv{a} = \begin{bmatrix} 1 \\ 3 \end{bmatrix} \quad \vv{b} = \begin{bmatrix} 0 & 2 \end{bmatrix} \quad \vv{c} = \begin{bmatrix} 2 \\ 0 \\ -1 \end{bmatrix} \end{split}\]

Compute the following or explain why it is undefined.

  1. \(2\vv{A}\)

    Solution
    \[\begin{split} 2\vv{A} = \begin{bmatrix} 2 \cdot 0 & 2 \cdot 2 \\ 2 \cdot 2 & 2 \cdot 4 \\ 2 \cdot 1 & 2 \cdot 3 \end{bmatrix} = \begin{bmatrix} 0 & 4 \\ 4 & 8 \\ 2 & 6 \end{bmatrix} \end{split}\]

  2. \(\vv{A + B}\)

    Solution

    Undefined because the shape of A (3×2) is different from the shape of B (2×3).

  3. \(\vv{A}^{\rm T} + \vv{B}\)

    Solution
    \[\begin{split} \vv{A}^{\rm T} + \vv{B} &= \begin{bmatrix} 0 & 2 & 1 \\ 2 & 4 & 3 \end{bmatrix} + \begin{bmatrix} 0 & 2 & 1\\ 2 & 4 & 3 \end{bmatrix} \\ &= \begin{bmatrix} 0+0 & 2+2 & 1+1 \\ 2+2 & 4+4 & 3+3 \end{bmatrix} \\ &= \begin{bmatrix} 0 & 4 & 2 \\ 4 & 8 & 6 \end{bmatrix} \end{split}\]

    (or, \(\vv{A}^{\rm T} = \vv{B}\) so \(\vv{A}^{\rm T} + \vv{B} = 2\vv{B}\).)

  4. \(\vv{C - D}\)

    Solution
    \[\begin{split} \vv{C} -\vv{D} &= \begin{bmatrix} 3-0 & 0-(-5) & 4-(-3) \\ -1-(-5) & 2-2 & 2-4 \\ 6-(-3) & 5-4 & -4-0 \end{bmatrix} \\ &= \begin{bmatrix} 3 & 5 & 7 \\ 4 & 0 & -2 \\ 9 & 1 & -4 \end{bmatrix} \end{split}\]

  5. \(\vv{Aa}\)

    Solution
    \[\begin{split} \vv{Aa} &= \begin{bmatrix} 0 \cdot 1 + 2 \cdot 3 \\ 2 \cdot 1 + 4 \cdot 3 \\ 1 \cdot 1 + 3 \cdot 3 \end{bmatrix} \\ &= \begin{bmatrix} 6 \\ 14 \\ 10 \end{bmatrix} \end{split}\]

  6. \(\vv{Ab}\)

    Solution

    Undefined because A has 2 columns but b has 1 row.

  7. \(\vv{Ac}\)

    Solution

    Undefined because A has 2 columns but c has 3 rows.

  8. \(\vv{Bc}\)

    Solution
    \[\begin{split} \vv{Bc} &= \begin{bmatrix} 0 \cdot 2 + 2 \cdot 0 + 1 \cdot -1 \\ 2 \cdot 2 + 4 \cdot 0 + 3 \cdot -1 \end{bmatrix} \\ &= \begin{bmatrix} -1 \\ 1 \end{bmatrix} \end{split}\]

  9. \(\vv{c}^{\rm T}\vv{A}\)

    Solution
    \[\begin{split} \vv{c}^{\rm T} \vv{A} &= \begin{bmatrix} 0 & 2 & -1\end{bmatrix} \begin{bmatrix} 0 & 2 \\ 2 & 4 \\ 1 &3 \end{bmatrix} \\ &= \begin{bmatrix} 2 \cdot 0 + 0 \cdot 2 + -1 \cdot 1 & 2 \cdot 2 + 0 \cdot 4 \cdot -1 \cdot 3 \end{bmatrix} \\ &= \begin{bmatrix} -1 & 1 \end{bmatrix} \end{split}\]
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