Julia cheatsheet¶

Version and Dependencies¶

This assumes

• Julia v1.0 or above

• using LinearAlgebra, Statistics, Compat has been run

• The while and for are assumed to be within a Jupyter Notebook or within a function. Otherwise, Julia 1.0 has different scoping rules for global variables, which will be made more consistent in a future release

Variables¶

Here are a few examples of basic kinds of variables we might be interested in creating.

Command

Description

A = 4.1
B = [1, 2, 3]
C = [1.1 2.2 3.3]
D = [1 2 3]'
E = [1 2; 3 4]


How to create a scalar, a vector, or a matrix. Here, each example will result in a slightly different form of output. A is a scalar, B is a flat array with 3 elements, C is a 1 by 3 vector, D is a 3 by 1 vector, and E is a 2 by 2 matrix.

s = "This is a string"


A string variable

x = true


A Boolean variable

Vectors and Matrices¶

These are a few kinds of special vectors/matrices we can create and some things we can do with them.

Command

Description

A = zeros(m, n)


Creates a matrix of all zeros of size m by n. We can also do the following:

A = zeros(B)


which will create a matrix of all zeros with the same dimensions as matrix or vector B.

A = ones(m, n)


Creates a matrix of all ones of size m by n. We can also do the following:

A = ones(B)


which will create a matrix of all ones with the same dimensions as matrix or vector B.

A = I


Creates a UniformScaling object which conforms to the dimensions required (e.g., in I +

zeros(2, 2), the I will act like 2 x 2.

A = j:k:n


This will create a sequence starting at j, ending at n, with difference k between points. For example, A = 2:4:10 will create the sequence 2, 6, 10 To convert the output to an array, use collect(A).

A = range(start, stop,
length = l)

A = range(start, stop,
step = s)


Creates a StepRangeLen iterable starting at start and ending at stop. Can be specified using either the length or step size (will not overshoot).

A = Diagonal(x)

Creates a Diagonal <: Matrix using the elements in x. For example if x = [1, 2, 3],

Diagonal(x) will return

$\begin{split}\begin{pmatrix} 1 & \cdot & \cdot\\ \cdot & 2 & \cdot \\ \cdot & \cdot & 3 \end{pmatrix}\end{split}$
A = rand(m, n)


Creates an m by n matrix of random numbers drawn from a uniform distribution on $$[0, 1]$$. Alternatively, rand can be used to draw random elements from a set X. For example, if X = [1, 2, 3], rand(X) will return either 1, 2, or 3.

A = randn(m, n)


Creates an m by n matrix of random numbers drawn from a standard normal distribution.

A[m, n]


This is the general syntax for accessing elements of an array or matrix, where m and n are integers. The example here returns the element in the second row and third column.

• We can also use ranges (like 1:3) in place of single numbers to extract multiple rows or columns

• A colon, :, by itself indicates all rows or columns

• The word end can also be used to indicate the last row or column

nrow, ncol = size(A)


Returns the number of rows and columns in a matrix. Alternatively, we can do

nrow = size(A, 1)


and

ncol = size(A, 2)

diag(A)


This function returns a vector of the diagonal elements of A (i.e., A[1, 1], A[2, 2], etc…).

A = hcat([1 2], [3 4])


Horizontally concatenates two matrices or vectors. The example here would return

$\begin{pmatrix} 1 & 2 & 3 & 4 \end{pmatrix}$

An alternative syntax is:

A = [[1 2] [3 4]]


For either of these commands to work, both matrices or vectors must have the same number of rows.

A = vcat([1 2], [3 4])


Vertically concatenates two matrices or vectors. The example here would return

$\begin{split}\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\end{split}$

An alternative syntax is:

A = [[1 2]; [3 4]]


For either of these commands to work, both matrices or vectors must have the same number of columns.

A = reshape(a, m, n)


Reshapes matrix or vector a into a new matrix or vector, A with m rows and n columns. For example A = reshape(1:10, 5, 2) would return

$\begin{split}\begin{pmatrix} 1 & 6 \\ 2 & 7 \\ 3 & 8 \\ 4 & 9 \\ 5 & 10 \end{pmatrix}\end{split}$

For this to work, the number of elements in a (number of rows times number of columns) must equal m * n.

A[:]


Converts matrix A to a vector. For example, if A = [1 2; 3 4], then A[:] will return

$\begin{split}\begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix}\end{split}$
reverse(A, dims = d)

Reverses the vector or matrix A along dimension d. For example, if A = [1 2 3; 4 5 6],

reverse(A, dims = 1)}, will reverse the rows of A and return

$\begin{split}\begin{pmatrix} 4 & 5 & 6 \\ 1 & 2 & 3 \end{pmatrix}\end{split}$

reverse(A, dims = 2) will reverse the columns of A and return

$\begin{split}\begin{pmatrix} 3 & 2 & 1 \\ 6 & 5 & 4 \end{pmatrix}\end{split}$
repeat(A, m, n)


Repeats matrix A, m times in the row direction and n in the column direction. For example, if A = [1 2; 3 4], repeat(A, 2, 3) will return

$\begin{split}\begin{pmatrix} 1 & 2 & 1 & 2 & 1 & 2 \\ 3 & 4 & 3 & 4 & 3 & 4 \\ 1 & 2 & 1 & 2 & 1 & 2 \\ 3 & 4 & 3 & 4 & 3 & 4 \end{pmatrix}\end{split}$

Mathematical Functions¶

Here, we cover some useful functions for doing math.

Command

Description

5 + 2
5 - 2
5 * 2
5 / 2
5 ^ 2
5 % 2


Scalar arithmetic operations: addition, subtraction, multiplication, division, power, remainder.

A .+ B
A .- B
A .* B
A ./ B
A .^ B
A .% B


Element-by-element operations on matrices. This syntax applies the operation element-wise to corresponding elements of the matrices.

More generally, the . notation is used for broadcasting, which iterates a function over a collection.

A * B


When A and B are matrices, * will perform matrix multiplication, as long as the number of columns in A is the same as the number of columns in B.

dot(A, B)
A ⋅ B


This function returns the dot product/inner product of the two vectors A and B. The two vectors need to be dimensionless or column vectors.

Can also be called with the unicode ⋅ (\cdot<TAB>)

transpose(A)


This syntax returns the transpose of the matrix A (i.e., reverses the dimensions of A).

For example if

$\begin{split}A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\end{split}$

then transpose(A) returns

$\begin{split}\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}\end{split}$

If A contains complex numbers

$\begin{split}A = \begin{pmatrix} 1-1i & 2+1i \\ 3-2i & 4+2i \end{pmatrix}\end{split}$

then transpose(A) returns

$\begin{split}\begin{pmatrix} 1-1i & 3-2i \\ 2+1i & 4+2i \end{pmatrix}\end{split}$

The function is recursive, so it will also transpose all elements if possible.

A'


This syntax returns the adjoint of the matrix A.

For example if A is a real matrix

$\begin{split}A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\end{split}$

then A' returns

$\begin{split}\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}\end{split}$

which is exactly the transpose.

If A contains complex numbers

$\begin{split}A = \begin{pmatrix} 1-1i & 2+1i \\ 3-2i & 4+2i \end{pmatrix}\end{split}$

then A' returns

$\begin{split}\begin{pmatrix} 1+1i & 3+2i \\ 2-1i & 4-2i \end{pmatrix}\end{split}$
sum(A)
maximum(A)
minimum(A)


These functions compute the sum, maximum, and minimum elements, respectively, in matrix or vector A. We can also add an additional argument for the dimension to compute the sum/maximum/minumum across. For example sum(A, dims = 2) will compute the row sums of A and maximum(A, dims =1) will compute the maxima of eachcolumn of A.

inv(A)


This function returns the inverse of the matrix A. Alternatively, we can do:

A ^ (-1)

det(A)


This function returns the determinant of the matrix A.

val, vec = eigen(A)


Returns the eigenvalues (val) and eigenvectors (vec) of matrix A. In the output, val[i] is the eigenvalue corresponding to eigenvector val[:, i].

norm(A)


Returns the Euclidean norm of matrix or vector A. We can also provide an argument p, like so:

norm(A, p)


which will compute the p-norm (the default p is 2). If A is a matrix, valid values of p are 1, 2 and Inf.

A \ b


If A is square, this syntax solves the linear system $$Ax = b$$. Therefore, it returns x such that A * x = b. If A is rectangular, it solves for the least-squares solution to the problem.

Programming¶

The following are useful basics for Julia programming.

Command

Description

# One line comment

#=
Comment block
=#


Two ways to make comments. Comments are useful for annotating code and explaining what it does. The first example limits your comment to one line and the second example allows the comments to span multiple lines between the #= and =#.

for i in iterable
# do something

end


A for loop is used to perform a sequence of commands for each element in an iterable object, such as an array. For example, the following for loop fills the vector l with the squares of the integers from 1 to 3:

N = 3
l = zeros(N, 1)
for i = 1:N
l[i] = i ^ 2
end

while i <= N
# do something
end


A while loop performs a sequence of commands as long as some condition is true. For example, the following while loop achieves the same result as the for loop above

l = [0]
while norm(l) < 5
push!(l, 2)
end

if i <= N
# do something
else
# do something else
end


An if/else statement performs commands if a condition is met. For example, the following squares x if x is 5, and cubes it otherwise:

if x == 5
x = x ^ 2
else
x = x ^ 3
end


We can also just have an if statement on its own, in which case it would square x if x is 5, and do nothing otherwise.

if x == 5
x = x ^ 2
end

fun(x, y) = 5 * x + y

function fun(x, y)
ret = 5 * x
return ret + y
end


These are two ways to define functions. Both examples here define equivalent functions.

The first method is for defining a function on one line. The name of the function is fun and it takes two inputs, x and y, which are specified between the parentheses. The code after the equals sign tells Julia what the output of the function is.

The second method is used to create functions of more than one line. The name of the function, fun, is specified right after function, and like the one-line version, has its arguments in parentheses. The return statement specifies the output of the function.

foo = x -> x + 3


Defines an anonymous function and binds it to the name foo.

println("Hello world")


Print to screen.

We can also print the values of variables to screen:

println("The value of x is \$(x).")