APPENDIX E

Linear Algebra Functions

This appendix will help readers translate the linear algebra they know into Julia code. This material is based on the official documentation for the LinearAlgebra¹ package and Eldén (2007).

E.1 Vector Operations

Some common vector operations in Julia are given in Table E.1.

Note that the Julia operations given in Table E.1 should not be considered an exclusive list. For example, both dot(x,y) and x·y are quoted for the inner product but ·(x,y) also works. The following code block illustrates some of the operations in Table E.1.

x = [1, 2]

y = [3, 4]

c = 8

c*x

# 2-element Array{Int64,1}:

# 8

# 16

dot(x,y)

# 11

x + c*ones(2)

# 2-element Array{Float64,1}:

# 9.0

# 10.0

normalize(x)

# 2-element Array{Float64,1}:

# 0.4472135954999579

# 0.8944271909999159

Table E.1 Common vector operations in Julia, using LinearAlgebra, for n-dimensional vectors x and y.

Operation	Notation	Julia
Addition	x + y	x+y or +(x, y)
Subtraction	x − y	x−y or −(x, y)
Scalar-vector addition	x + c1	x+c*ones(n)
Scalar-vector multiplication	cx	c*x or *(c, x)
Transpose	x′ or x⊤	x′ or transpose(x)
Inner product	x′y	dot(x,y) or x•y
Cross product	x × y	cross(x,y) or x×y
Norm	|| x ||	norm(x)
Distance	|| x − y ||	norm(x-y)
Normalize	x/|| x||	normalize(x)
Sum	${\displaystyle {\sum }_{i=1}^n{x}_i}$	sum(x)
Mean	$\left(1/n\right){\displaystyle {\sum }_{i=1}^n{x}_i}$	mean(x)

E.2 Matrix Operations

Some common matrix operations in Julia are given in Table E.2; again, this is not intended to be an exhaustive list.

The following code block illustrates some of the operations in Table E.2.

X = [5 1; 2 4]

Y = [2 0; 1 5]

y = [3, 4]

X*y

# 2-element Array{Int64,1}:

# 19

# 22

*(Y,X)

# 2x2 Array{Int64,2}:

# 10  2

# 15  21

## Row sums

sum(X,dims=1)

# 1x2 Array{Int64,2}:

# 7 5

Table E.2 Common matrix operations in Julia, where all operations assume compatible matrix and vector dimensions.

Operation	Notation	Julia
Addition	X + Y	X+Y or +(X,Y)
Subtraction	X − Y	X−Y or −(X,Y)
Transpose	X′ or X⊤	X′ or transpose(X)
Inverse	X−1	inv(X)
Moore-Penrose pseudo-inv.	X†	pinv(X)
Scalar-matrix multiplication	cX, c ∈ ℝ	*(c,X) or c*X
Vector-matrix multiplication	Xy	*(X,y) or X*y
Matrix-matrix multiplication	XY	*(X,Y) or X*Y
Matrix raised to a power p	Xp	X^p
Determinant	|X|	det(X)
Log-determinant	log(|X|)	logdet(X)
Log absolute value det.	log(||X||)	logabsdet(X)
Column sum	${\displaystyle {\sum }_{m\in M}{x}_{m,n}}$	sum(X, dims=1)
Row sum	${\displaystyle {\sum }_{n\in N}{x}_{m,n}}$	sum(X, dims=2)

E.3 Matrix Decompositions

Some common matrix decompositions in Julia are given in Table E.3.

The following code block illustrates how to use the decompositions in Table E.3.

using LinearAlgebra

X = [5 1 3; 0 8 2; 3 1 6]

Y = [8 0 1; 0 3 2; 1 2 5] # symmetric positive-definite

## Eigenvalue decomposition

evX=eigen(X)

# eigenvectors

evX.vectors

# 3x3 Array{Float64,2}:

# 0.7024 -0.384529 -0.497977

# 0.242773 0.7843 -0.652046

# -0.6691 -0.486837 -0.571712

# eigenvalues

evX.values

# 3-element Array{Float64,1}:

# 2.487860053322798

# 6.758543755579095

# 9.753596191098106

## Singular value decomposition

svdX=svd(X)

# matrix U

svdX.U

# 3x3 Array{Float64,2}:

# -0.476574 0.457265 -0.750857

# -0.655574 -0.753904 -0.0430237

# -0.585747 0.471739 0.659062

# matrix V

svdX.V

# 3x3 Adjoint{Float64,Array{Float64,2}}:

# -0.422445 0.539959 -0.728

# -0.643539 -0.744284 -0.178604

# -0.638278 0.393046 0.661904

# matrix Sigma

diagm(0 => svdX.S)

# 3x3 Array{Float64,2}:

# 9.80036 0.0 0.0

# 0.0 6.85522 0.0

# 0.0 0.0 2.44107

## Cholesky decomposition

cholY=cholesky(Y)

# matrix L

cholY.L

# 3x3 LowerTriangular{Float64,Array{Float64,2}}:

# 2.82843 . .

# 0.0 1.73205 .

# 0.353553 1.1547 1.88193

# note that cholY.U gives transpose(cholY.L)

## QR decomposition

qrX = qr(X)

# matrix Q

qrX.Q

# 3x3 LinearAlgebra.QRCompactWYQ{Float64,Array{Float64,2}}:

# -0.857493 0.0220386 -0.514024

# 0.0 -0.999082 -0.0428353

# -0.514496 -0.036731 0.856706

#matrix R

qrX.R

# 3x3 Array{Float64,2}:

# -5.83095 -1.37199 -5.65945

# 0.0 -8.00735 -2.15243

# 0.0 0.0 3.51249

Table E.3 Common matrix decompositions in Julia, where Notation is intended to help in understanding the subsequent code block.

Decomposition	Notation	Julia
Eigenvalue	Xm×m = PDP−1	eigen(X)
Singular value	Xm×n = Um×n∑m×n V′n×n, m = n	svd(X)
Cholesky	Xm×m = LL′, X positive definite	cholesky(X)
QR	Xm×n = Qm×mRm×n, m ≥n	qr(X)

¹http://docs.julialang.org/en/v1/stdlib/LinearAlgebra/index.html