MatrixRank

The rank of a matrix: the number of linearly independent rows (equivalently, columns).

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

MatrixRank(A)the rank of the matrix A.

SageMathmatrix(A).rank()SymPyMatrix(A).rank()

Typed signatures

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

Domain: Linear algebra

Details

• Written $\operatorname{rank} A$. Row rank equals column rank, so $\operatorname{rank} A = \operatorname{rank} A^\top$.
• A square matrix is invertible exactly when its rank is its size.
• Not the array Rank, which counts dimensions.

Examples

Basic #

live ↗

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

live ↗

In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack2, 4\bigr\rbrack\bigr\rbrack)$Out$1$

live ↗

In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack4, 5, 6\bigr\rbrack, \bigl\lbrack7, 8, 9\bigr\rbrack\bigr\rbrack)$Out$2$

Non-square matrices too

live ↗

In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack2, 4, 6\bigr\rbrack\bigr\rbrack)$Out$1$

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In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack0, 0\bigr\rbrack, \bigl\lbrack0, 0\bigr\rbrack\bigr\rbrack)$Out$0$

A matrix of pairwise-distinct symbols has full rank -- its determinant, as a polynomial in independent indeterminates, can never be identically zero

live ↗

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

Three rows, rank 3 despite the zeros

live ↗

In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack0, 5, 2, 4, 4\bigr\rbrack, \bigl\lbrack2, 5, 0, 4, 0\bigr\rbrack, \bigl\lbrack5, 1, 5, 4, 5\bigr\rbrack\bigr\rbrack)$Out$3$

live ↗

In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack a, b, c\bigr\rbrack, \bigl\lbrack d, m, f\bigr\rbrack, \bigl\lbrack g, h, k\bigr\rbrack\bigr\rbrack)$Out$3$

Scope #

Approximate entries

live ↗

In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack1, 2.5\bigr\rbrack, \bigl\lbrack3.5, 4\bigr\rbrack\bigr\rbrack)$Out$2$

Complex entries: the second row is $i$ times the first

live ↗

In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack1, i\bigr\rbrack, \bigl\lbrack i, -1\bigr\rbrack\bigr\rbrack)$Out$1$

Exact numeric entries

live ↗

In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack π, e\bigr\rbrack, \bigl\lbrack2π, 2e\bigr\rbrack\bigr\rbrack)$Out$1$

The second row is $\sqrt{2}$ times the first

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In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack\sqrt{2}, 1\bigr\rbrack, \bigl\lbrack2, \sqrt{2}\bigr\rbrack\bigr\rbrack)$Out$1$

A 3×4 matrix with one dependent row

live ↗

In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack1, 2, 3, 4\bigr\rbrack, \bigl\lbrack2, 4, 6, 8\bigr\rbrack, \bigl\lbrack1, 0, 1, 0\bigr\rbrack\bigr\rbrack)$Out$2$

live ↗

In$\operatorname{rank}(\mathrm{IdentityMatrix}(5))$Out$5$

A $3\times3$ matrix of pairwise-distinct symbols is full rank — the determinant, as a polynomial in independent indeterminates, can't be identically zero.

live ↗

In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack p, q, r\bigr\rbrack, \bigl\lbrack s, t, u\bigr\rbrack, \bigl\lbrack v, w, x\bigr\rbrack\bigr\rbrack)$Out$3$

Symbolic rows that are proportional have rank 1

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In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack2a, 2b\bigr\rbrack\bigr\rbrack)$Out$1$

The Hilbert matrix is invertible, so full rank

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In$\operatorname{rank}(\mathrm{HilbertMatrix}(3))$Out$3$

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In$\operatorname{rank}(\mathrm{IdentityMatrix}(3))$Out$3$

Applications #

The 4×4 Hilbert matrix is badly conditioned but invertible

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In$\operatorname{rank}(\mathrm{Tabulate}((\operatorname{\_1}, \operatorname{\_2})↦\frac{1}{(\operatorname{\_1}+\operatorname{\_2})-1}, 4, 4))$Out$4$

Properties #

Row rank equals column rank

live ↗

In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack4, 5, 6\bigr\rbrack, \bigl\lbrack7, 8, 9\bigr\rbrack\bigr\rbrack)=\operatorname{rank}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack4, 5, 6\bigr\rbrack, \bigl\lbrack7, 8, 9\bigr\rbrack\bigr\rbrack^T)$Out$\mathrm{True}$

Rank–nullity: the rank plus the dimension of the null space is the number of columns

live ↗

In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack4, 5, 6\bigr\rbrack, \bigl\lbrack7, 8, 9\bigr\rbrack\bigr\rbrack)+\mathrm{Length}(\ker\left(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack4, 5, 6\bigr\rbrack, \bigl\lbrack7, 8, 9\bigr\rbrack\bigr\rbrack\right))$Out$3$

An outer product $u v^\top$ has rank 1; a lazy Tabulate matrix is read as the matrix it describes

live ↗

In$\operatorname{rank}(\mathrm{Tabulate}(\mathrm{Multiply}, 4, 4))$Out$1$

Possible issues #

A symbolic matrix stays unevaluated (Wolfram answers the generic rank, 2)

live ↗

In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack x, 1\bigr\rbrack, \bigl\lbrack1, x\bigr\rbrack\bigr\rbrack)$Out$\operatorname{rank}(\bigl\lbrack\bigl\lbrack x, 1\bigr\rbrack, \bigl\lbrack1, x\bigr\rbrack\bigr\rbrack)$

A vector is treated as a 1×n matrix (Wolfram rejects it)

live ↗

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

A non-square matrix of distinct symbols doesn't qualify for the full-rank rule -- it stays unevaluated

live ↗

In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack a, b, c\bigr\rbrack, \bigl\lbrack d, g, h\bigr\rbrack\bigr\rbrack)$Out$\operatorname{rank}(\bigl\lbrack\bigl\lbrack a, b, c\bigr\rbrack, \bigl\lbrack d, g, h\bigr\rbrack\bigr\rbrack)$

Repeating a symbol across rows doesn't qualify either -- native compute-engine still answers this one directly (rank 1, equal rows)

live ↗

In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack a, b\bigr\rbrack\bigr\rbrack)$Out$1$

A zero row lowers the rank: only one independent row

live ↗

In$\operatorname{rank}(\bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack0, 0\bigr\rbrack\bigr\rbrack)$Out$1$

Neat examples #

Every matrix with entries $i + j$ has rank 2, whatever its size

live ↗

In$\operatorname{rank}(\mathrm{Tabulate}(\mathrm{Add}, 4, 4))$Out$2$

About

Examples	28
Checked against	Wolfram Language 24 agree3 differSageMath 19 agree2 unevaluatedSymPy 18 agree2 unevaluated
Crosswalk	Wolfram LanguageSageMathSymPy

Implementation

mappedsagepackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts