12.1: Creating Matrices and Arrays
- Page ID
- 85005
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)By Carey A. Smith
For our purposes, an array is 2-dimensional set of numbers.
A matrix is a 2-dimensional set of numbers that is used for linear algebra computations.
MATLAB keeps track of the numbers of rows and columns of a matrix.
User-defined matrices are typically defined in the following 2 equivalent ways:
Method 1: Create a matrix in directly 2-dimensions
M1 = [1 4 7 10
2 5 8 11
3 6 9 12]
Method 2: Create a matrix on 1 line, with semicolons separating the rows:
M2 = [1 4 7 10; 2 5 8 11; 3 6 9 12]
This creates the same matrices, but is not as visually obvious. This method is often used in books to save space on a page.
It is also possible to define MATLAB matrices (arrays) with 3 or more dimensions, but we will only use 2-dimensional arrays in this text.
Built-in functions to create a matrix
zeros(rows,cols)
Example:
z23 = zeros(2,3)
Result:
0 0 0
0 0 0
ones(rows,cols)
Example:
a32 = ones(3,2)
1 1
1 1
1 1
You can create a matrix of all constant values by multiplying a "ones" matrix by a constant.
6*ones(rows,cols)
a32_6 = 6*ones(3,2)
6 6
6 6
6 6
eye(n) = Identity matrix
Example:
I4 = eye(4)
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
magic(n) = Magic square. The sums of all the rows, columns, and diagonals are the same in a magic square.
Example:
m3 = magic(3)
8 1 6
3 5 7
4 9 2
diag(v) When v is a vector, creates a square matrix with diagonal elements = v.
v = [4, 9, 16]
B = diag(v)
4 0 0
0 9 0
0 0 16
diag(M) When M is a matrix, this extracts the diagonal elements of matrix M.
Example using magic square m3:
diag(m3) = [8
5
2]
pascal4 = pascal(4)
1 1 1 1
1 2 3 4
1 3 6 10
1 4 10 20
It has binomial coefficients
Matrix Transpose
Previously, we saw that a row vector can be transposed to change it to a column vector, like this:
a = [1 2 3]
at1 = a'
This becomes:
at1 =
1
2
3
You can also use the transpose( ) function:
at2 = transpose(a)
at2 =
1
2
3
The same function can be applied to any matrix, as shown by these examples:
Set
m3 = magic(3)
m3 = 8 1 6
3 5 7
4 9 2
m3t = m3'
m3t =
8 3 4
1 5 9
6 7 2
Set
m24 = [1 2
3 4
5 6
7 8]
m24t = m24'
m24t =
1 3 5 7
2 4 6 8