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TensorRank[tensor]

gives the rank of tensor.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Assumptions  
GenerateConditions  
Properties & Relations  
Possible Issues  
See Also
Tech Notes
Related Guides
Related Links
History
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TensorRank

TensorRank[tensor]

gives the rank of tensor.

Details and Options

  • TensorRank accepts any type of tensor, either symbolic or explicit, including any type of array.
  • On explicit rectangular arrays of scalars, TensorRank coincides with ArrayDepth. On symbolic arrays, TensorRank stays unevaluated unless the array has been assigned a rank through any form of assumption.
  • TensorRank takes the following options:
  • Assumptions $Assumptionsassumptions on parameters
    GenerateConditions Falsewhether to generate answers that involve conditions on parameters

Examples

open all close all

Basic Examples  (4)

Rank of an array:

Wolfram Language code: TensorRank[{{1, 3}, {2, Pi}}]

Rank of an array symbol:

Wolfram Language code: TensorRank[ArraySymbol["a", {p, q, r}]]

Rank of a tensor product of two tensors:

Wolfram Language code: TensorRank[TX]

Rank of a contraction:

Wolfram Language code: TensorRank[TensorContract[T, {{1, 3}}]]

Scope  (4)

Rank or depth of explicit arrays:

Wolfram Language code: A = Array[a, {2, 3, 4}]; TensorRank[A]
Wolfram Language code: A = SparseArray[{{1, 2, 3} -> a}, {2, 3, 4}]; TensorRank[A]
Wolfram Language code: A = SymmetrizedArray[pos_ :> a, {4, 4, 4}, Symmetric[All]]; TensorRank[A]

Rank of symbolic arrays:

Wolfram Language code: $Assumptions = { A∈Arrays[{4, 4, 5, 5}, Reals, Symmetric[{1, 2}]], M∈Matrices[{3, 4}, Reals], V∈Vectors[5, Reals] };
Wolfram Language code: TensorRank[A]
Wolfram Language code: TensorRank[M]
Wolfram Language code: TensorRank[V]

Rank of vector, matrix, and array symbols:

Wolfram Language code: TensorRank[VectorSymbol["v", n]]
Wolfram Language code: TensorRank[MatrixSymbol["m", {5, 7}]]
Wolfram Language code: TensorRank[ArraySymbol["a", {p, q, r, s}]]

Rank of general tensor expressions:

Wolfram Language code: TensorRank[abc]
Wolfram Language code: TensorRank[TensorContract[a, {{1, 5}, {2, 3}}]]
Wolfram Language code: TensorRank[TensorTranspose[a, {3, 4, 1, 2}]]

Options  (2)

Assumptions  (1)

Specify locally the domain of a symbolic array:

Wolfram Language code: TensorRank[AA, Assumptions -> A∈Arrays[{d, d}]]

GenerateConditions  (1)

By default, TensorRank quietly makes assumptions necessary for the input to be well-defined:

Wolfram Language code: a = MatrixSymbol["a", {m, n}]; b = MatrixSymbol["b", {p, q}]; TensorRank[a + b]

With GenerateConditionsTrue, TensorRank gives a conditional result:

Wolfram Language code: TensorRank[a + b, GenerateConditions -> True]

With GenerateConditionsNone, TensorRank fails when assumptions are necessary:

Wolfram Language code: TensorRank[a + b, GenerateConditions -> None]

Properties & Relations  (2)

On explicit arrays, TensorRank coincides with ArrayDepth:

Wolfram Language code: A = Array[a, {2, 3, 4}]; {TensorRank[A], ArrayDepth[A]}
Wolfram Language code: A = SparseArray[{{1, 2, 3} -> a}, {2, 3, 4}]; {TensorRank[A], ArrayDepth[A]}
Wolfram Language code: A = SymmetrizedArray[pos_ :> a, {4, 4, 4}, Symmetric[All]]; {TensorRank[A], ArrayDepth[A]}

For symbolic expressions, there is no default rank assumed:

Wolfram Language code: TensorRank[A]

Use assumptions to assign a rank to the array:

Wolfram Language code: Assuming[A∈Arrays[{3, 4, d, d}], TensorRank[A]]
Wolfram Language code: $Assumptions = A∈Arrays[{3, 4, 5}]; TensorRank[A]

Possible Issues  (3)

TensorRank can obtain some information contextually. Expressions without tensor properties inside numeric functions, arrays, or derivatives are considered scalars:

Wolfram Language code: TensorRank[x]
Wolfram Language code: TensorRank[1 / x]
Wolfram Language code: TensorRank[{x}]
Wolfram Language code: Grad[x, {a, b}]

It is not possible to mix incompatible local and global assumptions:

Wolfram Language code: $Assumptions = A∈Arrays[{2, 2, 2, 2}];
Wolfram Language code: Assuming[A∈Arrays[{2, 2}], TensorRank[A]]

TensorRank does not check for dimensions homogeneity, only rank homogeneity:

Wolfram Language code: $Assumptions = {v∈Vectors[2], w∈Vectors[3]};
Wolfram Language code: TensorRank[v + w]
Wolfram Language code: TensorDimensions[v + w]

With GenerateConditionsTrue, TensorRank checks for dimensions homogeneity:

Wolfram Language code: TensorDimensions[v + w, GenerateConditions -> True]
Wolfram Research (2012), TensorRank, Wolfram Language function, https://reference.wolfram.com/language/ref/TensorRank.html (updated 2024).

Text

Wolfram Research (2012), TensorRank, Wolfram Language function, https://reference.wolfram.com/language/ref/TensorRank.html (updated 2024).

CMS

Wolfram Language. 2012. "TensorRank." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/TensorRank.html.

APA

Wolfram Language. (2012). TensorRank. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TensorRank.html

BibTeX

@misc{reference.wolfram_2026_tensorrank, author="Wolfram Research", title="{TensorRank}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/TensorRank.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_tensorrank, organization={Wolfram Research}, title={TensorRank}, year={2024}, url={https://reference.wolfram.com/language/ref/TensorRank.html}, note=[Accessed: 13-June-2026]}