3.10: Element-by-Element Array Operations
- Page ID
- 134993
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Scalar operations involve one array and one scalar. But many engineering problems require operations between two arrays.
For example, suppose we have two arrays:
a = [-1 3 2 10];
b = [5 1 8 3];
If we add the arrays:
c = a + b
MATLAB adds corresponding elements and returns:
c = [4 4 10 13]
This means:
c = [a(1)+b(1), a(2)+b(2), a(3)+b(3), a(4)+b(4)]
or:
c = [-1+5, 3+1, 2+8, 10+3]
Subtraction works the same way:
d = a - b
MATLAB subtracts each element of b from the corresponding element of a.
The result is:
d = [-6 2 -6 7]
Multiplication, division, and powers work similarly, but we place a dot to the left of the operation. For example, to multiply a and b:
e = a .* b
the result is:
e = [-5 3 16 30]
The table below summarizes the element-by-element operations between two arrays a and b.
|
Operation |
MATLAB form |
Meaning |
|
Addition |
a + b |
Add corresponding elements |
|
Subtraction |
a - b |
Subtract corresponding elements |
|
Element-wise multiplication |
a .* b |
Multiply corresponding elements |
|
Element-wise division |
a ./ b |
Divide corresponding elements |
|
Element-wise exponentiation |
a .^ b |
Raise each element of a to the matching power in b |
Example: Calculating Kinetic Energy
In physics and engineering, the kinetic energy of an object is:
KE = 0.5 * m * v^2
Suppose the mass is constant, but we measure several different velocities:
m = 2.5;
v = [0 3 6 9 12];
To calculate kinetic energy for every velocity, we write:
KE = 0.5 * m * v.^2
The .^2 is required because we want to square every velocity value individually.
The result is:
KE = [0 11.2500 45.0000 101.2500 180.0000]
This one line calculates the kinetic energy at all five velocities.
When performing element-by-element operations between two arrays, the arrays usually need to have the same size.
Common Beginner Mistake
A very common beginner mistake is forgetting the dot in multiplication, division, or exponentiation.
For example, suppose we want to square every value in an array:
x = [1 2 3 4];
The correct command is:
xSquared = x.^2
not:
xSquared = x^2
The first command squares each element. The second command asks MATLAB to perform matrix exponentiation, which is a different mathematical operation.