Wolfram: Nested List, Array, Tree, Dimensions

Nested List: Array, Matrix, Tree

List is critically important in WolframLang. Here's some notion on tree, array, you need to understand.

Matrix (aka rectangular array, n-dimensional array)

A nested list where there are same number of elements at each level of nesting. e.g. {{3,1}, {4, 5}, {2, 4}}.

Tree

(aka jagged array) A arbitrarily nested list. e.g. {3, {4, 5}, {9}}.

Dimensions and ArrayDepth

Dimensions[expr]

Shape of rectangular array. A list of integers that is the count at each level of nesting.

e.g. a 3 by 2 matrix has dimensions {3,2}.

• Dimensions

Dimensions[{a, b}]
(* {2} *)

Dimensions[{x, y, z}]
(* {3} *)

Dimensions[{{a, b}, {c, d}}]
(* {2, 2} *)

Dimensions[{{{a}, {b}}, {{c}, {d}}}]
(* {2, 2, 1} *)

(* jagged array. return the max regularity *)
Dimensions[{{{a}, {b}}, {{c}, d}}]
(* {2, 2} *)

ArrayDepth[expr]

same as Length[Dimensions[expr]]

typically known as the “dimension” of a array.

• ArrayDepth

(* 2 by 2 matrix *)
Dimensions[{{a,b},{c,d}}]
(* {2, 2} *)

ArrayDepth[{{a,b},{c,d}}]
(* 2 *)

(* s------------------------------ *)

(* 2 by 3 matrix *)
Dimensions[{{a,b,c},{d,e,f}}]
(* {2, 3} *)

ArrayDepth[{{a,b,c},{d,e,f}}]
(* 2 *)

(* s------------------------------ *)

(* creat a 2 by 2 by 2 matrix *)
Table[ f[x,y,z], {x,1,2},{y,1,2},{z,1,2}]
(*
{
{
 {f[1, 1, 1], f[1, 1, 2]},
 {f[1, 2, 1], f[1, 2, 2]}},
{
 {f[2, 1, 1], f[2, 1, 2]},
 {f[2, 2, 1], f[2, 2, 2]}
}
}
*)

Dimensions @ Table[ f[x,y,z], {x,1,2},{y,1,2},{z,1,2}]
(* {2, 2, 2} *)

ArrayDepth @ Table[ f[x,y,z], {x,1,2},{y,1,2},{z,1,2}]
(* 3 *)