Rank

The number of dimensions of an array: 0 for a scalar, 1 for a vector, 2 for a matrix.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageArrayDepth!↗

Rank(x)the length of the shape of x.

Typed signatures

Type	Package	Applies
(value) -> integer	combinatoricsreplaces compute-engine	always
(value) -> integer	compute-engine	always

Domain: Linear algebra

Details

• Tensor rank in the array sense — Wolfram's ArrayDepth, NumPy's ndim.
• Not the rank of a matrix (its number of independent rows); that is MatrixRank.

Examples

Basic #

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In$\mathrm{Rank}(5)$Out$0$

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In$\mathrm{Rank}(\bigl\lbrack1, 2, 3\bigr\rbrack)$Out$1$

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In$\mathrm{Rank}(\bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack)$Out$2$

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In$\mathrm{Rank}(\bigl\lbrack\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack5, 6\bigr\rbrack, \bigl\lbrack7, 8\bigr\rbrack\bigr\rbrack\bigr\rbrack)$Out$3$

Symbolic entries

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In$\mathrm{Rank}(\bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c, d\bigr\rbrack\bigr\rbrack)$Out$2$

A symbol is a scalar

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In$\mathrm{Rank}(x)$Out$0$

Scope #

Rectangular arrays

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In$\mathrm{Rank}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack4, 5, 6\bigr\rbrack\bigr\rbrack)$Out$2$

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In$\mathrm{Rank}(\mathrm{IdentityMatrix}(3))$Out$2$

A ragged list has depth 1 -- only its outer level is a full array

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In$\mathrm{Rank}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3\bigr\rbrack\bigr\rbrack)$Out$1$

...likewise a list mixing scalars and lists

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In$\mathrm{Rank}(\bigl\lbrack1, \bigl\lbrack2, 3\bigr\rbrack\bigr\rbrack)$Out$1$

The empty list is still a vector, of length 0

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In$\mathrm{Rank}(\bigl\lbrack\bigr\rbrack)$Out$1$

...and $\{\{\}\}$ a 1×0 matrix

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In$\mathrm{Rank}(\bigl\lbrack\bigl\lbrack\bigr\rbrack\bigr\rbrack)$Out$2$

A 2×3 grid of unevaluated calls still reads as a matrix, from its dimensions alone

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In$\mathrm{Rank}(\mathrm{Tabulate}(f, 2, 3))$Out$2$

A ragged list is only a vector of lists, so rank $1$

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In$\mathrm{Rank}(\bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c\bigr\rbrack\bigr\rbrack)$Out$1$

Properties #

The length of the array's shape

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In$\mathrm{Rank}(\bigl\lbrack\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack5, 6\bigr\rbrack, \bigl\lbrack7, 8\bigr\rbrack\bigr\rbrack\bigr\rbrack)=\mathrm{Length}(\mathrm{Shape}(\bigl\lbrack\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack, \bigl\lbrack\bigl\lbrack5, 6\bigr\rbrack, \bigl\lbrack7, 8\bigr\rbrack\bigr\rbrack\bigr\rbrack))$Out$\mathrm{True}$

Rank equals the length of the shape

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In$\mathrm{Rank}(\bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c, d\bigr\rbrack\bigr\rbrack)=\mathrm{Length}(\mathrm{Shape}(\bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack c, d\bigr\rbrack\bigr\rbrack))$Out$\mathrm{True}$

A scalar has rank $0$

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In$\mathrm{Rank}(1)$Out$0$

A flat list has rank $1$

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In$\mathrm{Rank}(\bigl\lbrack1, 2\bigr\rbrack)$Out$1$

One nested level gives rank $2$

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In$\mathrm{Rank}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack\bigr\rbrack)$Out$2$

The rank of an array equals the number of dimensions given

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In$\mathrm{Rank}(\mathrm{Array}(a, \bigl\lbrack2, 1, 4, 3\bigr\rbrack))=\mathrm{Length}(\bigl\lbrack2, 1, 4, 3\bigr\rbrack)$Out$\mathrm{True}$

Possible issues #

A singular 2×2 matrix still has Rank 2; its MatrixRank is 1

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In$\mathrm{Rank}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack2, 4\bigr\rbrack\bigr\rbrack)$Out$2$

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In$\mathrm{Rank}(\sqrt{2})$Out$0$

About

Examples	22
Checked against	Wolfram Language 21 agree1 differ
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