Matrix Rank: Definition & Computation

What Rank Measures

The rank of an

m \times n

matrix

A

is a single non-negative integer

r

that captures how much of the matrix's potential dimensionality is actually used. It satisfies

Rank Bounds

0 \leq \text{rank}(A) \leq \min(m, n)

Rank bounds →

When

\text{rank}(A) = \min(m, n)

, the matrix has full rank — every row and every column contributes something that no combination of the others can reproduce. When

\text{rank}(A) < \min(m, n)

, the matrix is rank-deficient, meaning at least one row or column is a linear combination of the others.

A

5 \times 3

matrix with rank

3

uses all three of its column directions. A

5 \times 3

matrix with rank

2

has one column that is redundant — it lies in the span of the other two. The rank does not say which column is redundant (often more than one subset works), only that the effective column count is

2

.

Rank Notation

Notation

Rank Notation

One unqualified word standing for two quantities that are only equal because a theorem says so, an operator with four spellings, and a bound whose ceiling names the two conditions a matrix can be in. All of them are catalogued among the linear algebra symbols.

The

m \times n

shape and the entry brackets come from matrix operation notation;

\dim

,

\operatorname{Col}

and

\operatorname{Nul}

from fundamental-space notation;

\ker

,

\operatorname{Im}

and the rank–nullity dictionary from image and kernel notation.

\operatorname{rank}(A)

the rank of A

An operator name set upright, like

\det

and

\operatorname{tr}

— it takes a matrix of any shape and returns a single non-negative integer, as What Rank Measures above defines. Unlike the trace and the determinant it does not require a square matrix, which is why the bound below has to mention both dimensions.

Also written

\operatorname{rk}(A)

is the common short form and appears on this page alongside the long one;

\operatorname{rg}(A)

in German and French texts (from Rang); and

\rho(A)

in older and some engineering sources — a spelling worth knowing but not imitating, since

\rho

is heavily booked elsewhere.

Do not confuseThe Greek

\rho

at its other posts — spectral radius in numerical analysis, correlation in statistics, density in physics. When a source writes

\rho(A)

for a matrix, the surrounding claim decides whether it means rank or spectral radius, and the two are unrelated numbers.

0 \le \operatorname{rank}(A) \le \min(m, n)

rank runs from zero up to the smaller dimension

The ceiling is

\min(m, n)

, not

m

or

n

— the shorter side of the matrix caps how many independent directions there can be. Hitting the ceiling is full rank; falling short is rank-deficient. Those two words are the notation's whole vocabulary for a matrix's condition.

CasesOnly the zero matrix reaches the floor:

\operatorname{rank}(A) = 0

means every entry vanishes. For a square

n \times n

matrix, full rank

= n

is exactly invertibility, which is why

\operatorname{rank}(A) = n

,

\det(A) \neq 0

and "

A^{-1}

exists" are three spellings of one condition — the determinant and inverse pages state the same fact in their own marks.

Do not confuseReading "full rank" as "square". A

5 \times 3

matrix of rank

3

is at full rank while being nowhere near square — the term measures attainment of the

\min(m,n)

ceiling, not the shape of the array.

column rank

=

row rank

= \operatorname{rank}(A)

the two rank counts coincide, so the bare word is unambiguous

The single unqualified word is licensed by a theorem, not by convention: the columns live in

\mathbb{R}^m

and the rows in

\mathbb{R}^n

— different spaces — yet the two counts always agree, as Column Rank and Row Rank below proves. Only that equality makes it safe to write

\operatorname{rank}(A)

with no qualifier at all.

CasesThe qualified forms survive where the distinction is being argued rather than used — a proof will say "row rank" while establishing the equality, then drop the adjective forever after. In the four fundamental subspaces the same number appears as

\dim(\operatorname{Col} A)

and

\dim(\operatorname{Row} A)

, which is the dimension dialect of the same claim.

Do not confuseTreating the equality as a definition. It is a result — the notation quietly depends on it, and a reader who assumes the bare word was simply defined to be unambiguous misses that something had to be proved before the word could exist.

All linear algebra symbols →↑ Matrices

Column Rank and Row Rank

The column rank of

A

is the dimension of its column space — the subspace of

\mathbb{R}^m

spanned by the columns of

A

. It counts the maximum number of linearly independent columns.

The row rank is the dimension of the row space — the subspace of

\mathbb{R}^n

spanned by the rows. It counts the maximum number of linearly independent rows.

A fundamental theorem states that these two numbers are always equal:

\text{column rank of } A = \text{row rank of } A

This common value is called the rank of

A

, written

\text{rank}(A)

or

\text{rk}(A)

.

The equality is not obvious. The columns live in

\mathbb{R}^m

and the rows live in

\mathbb{R}^n

— two different spaces, potentially of different dimensions. The proof goes through row reduction: elementary row operations do not change the row space, and in reduced row echelon form, the number of nonzero rows (row rank) equals the number of pivot columns (column rank). Since row operations preserve both counts, the equality holds for the original matrix.

Computing Rank via Row Reduction

The standard method for finding the rank of a matrix is to reduce it to row echelon form and count the pivot positions.

Worked Example

A = \begin{pmatrix} 1 & 2 & 0 & -1 & 3 \\ 2 & 4 & 1 & 0 & 5 \\ -1 & -2 & 3 & 4 & 1 \\ 0 & 0 & 2 & 3 & -1 \end{pmatrix}

Subtract

2

times row

1

from row

2

, and add row

1

to row

3

:

\begin{pmatrix} 1 & 2 & 0 & -1 & 3 \\ 0 & 0 & 1 & 2 & -1 \\ 0 & 0 & 3 & 3 & 4 \\ 0 & 0 & 2 & 3 & -1 \end{pmatrix}

Subtract

3

times row

2

from row

3

, and subtract

2

times row

2

from row

4

:

\begin{pmatrix} 1 & 2 & 0 & -1 & 3 \\ 0 & 0 & 1 & 2 & -1 \\ 0 & 0 & 0 & -3 & 7 \\ 0 & 0 & 0 & -1 & 1 \end{pmatrix}

Subtract

3

times row

4

from row

3

:

\begin{pmatrix} 1 & 2 & 0 & -1 & 3 \\ 0 & 0 & 1 & 2 & -1 \\ 0 & 0 & 0 & 0 & 4 \\ 0 & 0 & 0 & -1 & 1 \end{pmatrix}

Swap rows

3

and

4

to place the pivot:

\begin{pmatrix} 1 & 2 & 0 & -1 & 3 \\ 0 & 0 & 1 & 2 & -1 \\ 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & 0 & 4 \end{pmatrix}

There are four pivots, in columns

1

,

3

,

4

, and

5

. So

\text{rank}(A) = 4

. Column

2

is the only non-pivot column, corresponding to the single free variable if this matrix were the coefficient matrix of a system.

Elimination finished, pivots counted

The pivots have been marked and counted, and that count is the rank. Row operations were applied throughout without changing the answer, because none of them can create or destroy an independent direction — they only make the existing ones easy to see. Run the elimination on a matrix of your own on the rank visualizer.

Counting pivots is therefore not an approximation to the rank but a direct measurement of it.

Rank and Dimension

For an

m \times n

matrix

A

, the rank can be at most

\min(m, n)

. Whether it reaches this maximum depends on the matrix's entries, not just its shape.

Full column rank means

\text{rank}(A) = n

— all

n

columns are independent. When

A

has full column rank, the system

Ax = \mathbf{b}

has at most one solution for any

\mathbf{b}

, because no free variables exist. The null space is

\{\mathbf{0}\}

.

Full row rank means

\text{rank}(A) = m

— all

m

rows are independent. When

A

has full row rank, the system

Ax = \mathbf{b}

has at least one solution for every

\mathbf{b}

, because the column space is all of

\mathbb{R}^m

.

When

A

is square (

m = n

) and has full rank

n

, both conditions hold simultaneously: the system has exactly one solution for every right-hand side, and

A

is invertible.

Rank and Linear Systems

The solvability of a linear system

Ax = \mathbf{b}

is determined entirely by comparing the rank of the coefficient matrix

A

with the rank of the augmented matrix

[A \mid \mathbf{b}]

.

A solution exists if and only if

\text{rank}(A) = \text{rank}([A \mid \mathbf{b}])

. This condition means that

\mathbf{b}

lies in the column space of

A

— it can be expressed as a linear combination of the columns.

When solutions exist, uniqueness depends on whether the rank equals the number of unknowns

n

. If

\text{rank}(A) = n

, there are no free variables and the solution is unique. If

\text{rank}(A) < n

, there are

n - \text{rank}(A)

free variables, and the solution set is an infinite family parametrized by those free variables.

The three possible outcomes are:

\text{rank}(A) < \text{rank}([A \mid \mathbf{b}])

means the system is inconsistent and has no solution.

\text{rank}(A) = \text{rank}([A \mid \mathbf{b}]) = n

means there is exactly one solution.

\text{rank}(A) = \text{rank}([A \mid \mathbf{b}]) < n

means there are infinitely many solutions.

There is no scenario with a finite number of solutions greater than one. A linear system either has zero, one, or infinitely many solutions.

Outcome	Rank condition	Number of solutions
Inconsistent	rank(A) < rank([A \| b])	0 (no solution)
Unique solution	rank(A) = rank([A \| b]) = n	exactly 1
Infinite solutions	rank(A) = rank([A \| b]) < n; n − rank(A) free variables	∞ (parametrized family)

The Rank-Nullity Theorem

For an

m \times n

matrix

A

, the rank and the nullity — the dimension of the null space

\{\mathbf{x} : A\mathbf{x} = \mathbf{0}\}

— satisfy

\text{rank}(A) + \text{nullity}(A) = n

The

n

columns of

A

partition into two groups: the pivot columns, which contribute to the column space and drive the rank, and the free columns, which contribute to the null space and drive the nullity. Every column does exactly one of these things.

For a

3 \times 5

matrix with rank

2

, the nullity is

3

. The column space is a two-dimensional subspace of

\mathbb{R}^3

(a plane through the origin), and the null space is a three-dimensional subspace of

\mathbb{R}^5

.

For a square

n \times n

matrix, the theorem says

\text{rank}(A) + \text{nullity}(A) = n

. If the rank is

n

(full rank), the nullity is

0

— the null space contains only

\mathbf{0}

, and

A

is invertible. If the rank is less than

n

, the null space is nontrivial, the determinant is zero, and

A

is singular.

The rank-nullity theorem is sometimes called the dimension theorem for linear maps. If

A

defines a linear transformation

T: \mathbb{R}^n \to \mathbb{R}^m

, then the rank is the dimension of the image (range) of

T

, and the nullity is the dimension of the kernel. Their sum equals the dimension of the domain.

Properties of Rank

The rank function obeys several inequalities and identities that constrain how matrix operations affect it.

The rank of the zero matrix is

0

, and this is the only matrix with rank zero. For any nonzero scalar

c

,

\text{rank}(cA) = \text{rank}(A)

— scaling does not create or destroy independence.

Transposition preserves rank:

Rank of Transpose

\text{rank}(A^T) = \text{rank}(A)

Rank of transpose →

This is a restatement of the equality of row rank and column rank.

The rank of a product can only decrease:

Rank Product Inequality

\text{rank}(AB) \leq \min(\text{rank}(A), \text{rank}(B))

Rank product inequality →

Multiplying by a matrix can collapse dimensions but cannot create new independent directions. There is also a lower bound due to Sylvester's inequality:

Sylvester Rank Inequality

\text{rank}(A) + \text{rank}(B) - n \leq \text{rank}(AB)

Sylvester rank inequality →

for

A

of size

m \times n

and

B

of size

n \times p

. This says the rank of the product cannot drop too far below the ranks of the factors.

The rank of a sum is bounded by the sum of the ranks:

Rank Sum Inequality

\text{rank}(A + B) \leq \text{rank}(A) + \text{rank}(B)

Rank sum inequality →

Equality holds when the column spaces of

A

and

B

are disjoint (intersect only at

\mathbf{0}

).

Multiplying by an invertible matrix preserves rank exactly:

Rank Invariance Invertible

\text{rank}(PAQ) = \text{rank}(A)

Rank invariance invertible →

for invertible

P

and

Q

. This is because invertible matrices neither collapse nor create dimensions.

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Properties of rank

Rank is constrained rather than computed by these rules — most are inequalities, not identities. That asymmetry is the point: operations can destroy independence but never create it.

08properties

Unconditional3

i

Rank bounds§ 1

0 \leq \operatorname{rank}(A) \leq \min(m, n)

Rank cannot exceed either dimension, because it counts independent rows and independent columns…

Unconditional

Requires: $A$ is $m \times n$

Rank cannot exceed either dimension, because it counts independent rows and independent columns simultaneously and those counts agree. A matrix reaching this bound has full rank; anything less is rank-deficient, meaning some row or column is a linear combination of the others.

ii

Zero matrix§ 7

\operatorname{rank}(A) = 0 \iff A = O

The only matrix of rank zero.

Unconditional

The only matrix of rank zero. Any nonzero entry gives at least one independent row, so rank zero and the zero matrix are the same condition stated two ways.

iv

Transpose invariance§ 2

\operatorname{rank}(A^{\mathsf{T}}) = \operatorname{rank}(A)

A restatement of row rank equalling column rank.

Unconditional

Requires: any $A$

A restatement of row rank equalling column rank. It is the reason rank can be defined without saying which of the two is meant, and why a wide matrix and its tall transpose describe the same amount of structure.

Conditional5

iii

Scalar multiple§ 7

\operatorname{rank}(cA) = \operatorname{rank}(A)

Scaling neither creates nor destroys independence.

Conditional

Holds when: $c \neq 0$

Scaling neither creates nor destroys independence. At $c = 0$ the statement collapses — every rank drops to zero at once — which is why the condition is not decoration.

v

Rank of a product§ 7

\operatorname{rank}(AB) \leq \min\bigl(\operatorname{rank}(A), \operatorname{rank}(B)\bigr)

Multiplication can collapse directions but never create independent ones — the image of AB sits…

Upper bound only

Holds when: product defined

Multiplication can collapse directions but never create independent ones — the image of $AB$ sits inside the image of $A$ , and its rank is capped by whatever $B$ passes through first. Equality is common but not guaranteed, so this bounds rather than determines.

Strict inequalityA = [[1, 0], [0, 0]], B = [[0, 0], [0, 1]]
rank(A) = rank(B) = 1, but AB = O with rank 0

vi

Sylvester’s inequality§ 7

\operatorname{rank}(A) + \operatorname{rank}(B) - n \leq \operatorname{rank}(AB)

The companion to the bound above: rank cannot drop arbitrarily far.

Lower bound

Holds when: $A$ is $m \times n$ and $B$ is $n \times p$

The companion to the bound above: rank cannot drop arbitrarily far. If both factors have full rank $n$ , the inequality forces $\operatorname{rank}(AB) \geq n$ , so the product has full rank too — which is how the invertibility of a product of invertible matrices follows.

vii

Rank of a sum§ 7

\operatorname{rank}(A + B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)

Adding matrices cannot produce more independent directions than the two contribute between…

Upper bound only

Holds when: $A, B$ same shape
Common error: reading it as an equality — the sum can have lower rank than either factor

Adding matrices cannot produce more independent directions than the two contribute between them, but it can produce fewer. Unlike the determinant, which has no sum rule at all, rank has a bound — just not an identity.

WitnessA = I₂, B = −I₂, both rank 2
A + B = O with rank 0, far below either

viii

Multiplication by an invertible matrix§ 7

\operatorname{rank}(PAQ) = \operatorname{rank}(A)

An invertible matrix loses nothing, so it cannot change the rank of what it multiplies.

Conditional

Holds when: $P$ and $Q$ invertible

An invertible matrix loses nothing, so it cannot change the rank of what it multiplies. This is why row reduction computes rank correctly — every elementary operation is multiplication by an invertible matrix, and the echelon form has the same rank as the original.

Unconditional

Conditional

Rank bounds§ 1Unconditional

0 \leq \operatorname{rank}(A) \leq \min(m, n)

Requires: $A$ is $m \times n$

Rank cannot exceed either dimension, because it counts independent rows and independent columns simultaneously and those counts agree. A matrix reaching this bound has full rank; anything less is rank-deficient, meaning some row or column is a linear combination of the others.

Read the full section

Zero matrix§ 7Unconditional

\operatorname{rank}(A) = 0 \iff A = O

The only matrix of rank zero. Any nonzero entry gives at least one independent row, so rank zero and the zero matrix are the same condition stated two ways.

Read the full section

Scalar multiple§ 7Conditional

\operatorname{rank}(cA) = \operatorname{rank}(A)

Holds when: $c \neq 0$

Scaling neither creates nor destroys independence. At

c = 0

the statement collapses — every rank drops to zero at once — which is why the condition is not decoration.

Read the full section

Transpose invariance§ 2Unconditional

\operatorname{rank}(A^{\mathsf{T}}) = \operatorname{rank}(A)

Requires: any $A$

A restatement of row rank equalling column rank. It is the reason rank can be defined without saying which of the two is meant, and why a wide matrix and its tall transpose describe the same amount of structure.

Read the full section

Rank of a product§ 7Upper bound only

\operatorname{rank}(AB) \leq \min\bigl(\operatorname{rank}(A), \operatorname{rank}(B)\bigr)

Holds when: product defined

Multiplication can collapse directions but never create independent ones — the image of

AB

sits inside the image of

A

, and its rank is capped by whatever

B

passes through first. Equality is common but not guaranteed, so this bounds rather than determines.

Strict inequalityA = [[1, 0], [0, 0]], B = [[0, 0], [0, 1]]
rank(A) = rank(B) = 1, but AB = O with rank 0

Read the full section

Sylvester’s inequality§ 7Lower bound

\operatorname{rank}(A) + \operatorname{rank}(B) - n \leq \operatorname{rank}(AB)

Holds when: $A$ is $m \times n$ and $B$ is $n \times p$

The companion to the bound above: rank cannot drop arbitrarily far. If both factors have full rank

n

, the inequality forces

\operatorname{rank}(AB) \geq n

, so the product has full rank too — which is how the invertibility of a product of invertible matrices follows.

Read the full section

Rank of a sum§ 7Upper bound only

\operatorname{rank}(A + B) \leq \operatorname{rank}(A) + \operatorname{rank}(B)

Holds when: $A, B$ same shape
Common error: reading it as an equality — the sum can have lower rank than either factor

Adding matrices cannot produce more independent directions than the two contribute between them, but it can produce fewer. Unlike the determinant, which has no sum rule at all, rank has a bound — just not an identity.

WitnessA = I₂, B = −I₂, both rank 2
A + B = O with rank 0, far below either

Read the full section

Multiplication by an invertible matrix§ 7Conditional

\operatorname{rank}(PAQ) = \operatorname{rank}(A)

Holds when: $P$ and $Q$ invertible

An invertible matrix loses nothing, so it cannot change the rank of what it multiplies. This is why row reduction computes rank correctly — every elementary operation is multiplication by an invertible matrix, and the echelon form has the same rank as the original.

Read the full section

The one-directional pattern runs through every entry. Multiplication can only lose rank, addition can only be bounded above, and the sole exact identities are the ones where nothing is lost at all — transposition, scaling, and multiplication by an invertible matrix.

Properties of rank·/linear-algebra/matrix/rankLearn Math Class

Rank of Special Matrices

Several matrix types have rank that can be read off directly from their structure.

The identity matrix

I_n

has rank

n

— all columns are standard basis vectors, which are linearly independent. Every invertible matrix has full rank by definition.

A diagonal matrix has rank equal to the number of nonzero diagonal entries. The zero entries correspond to collapsed coordinate directions.

A rank-

1

matrix has the form of an outer product of two nonzero vectors:

Rank-One Outer Product

A = \mathbf{u}\mathbf{v}^T, \quad \text{rank}(A) = 1

Rank one outer product →

Every column of

A

is a scalar multiple of

\mathbf{u}

, so the column space is the one-dimensional line through

\mathbf{u}

. Equivalently, every row is a scalar multiple of

\mathbf{v}^T

. Rank-

1

matrices are the building blocks of the outer product decomposition of matrix multiplication.

A symmetric positive definite matrix always has full rank — all its eigenvalues are strictly positive, so no dimension is collapsed. A nilpotent matrix of order

n > 1

always has rank strictly less than

n

, since its determinant is zero.

The rank of

A^T A

equals the rank of

A

:

Gram Rank Identity

\text{rank}(A^T A) = \text{rank}(A)

Gram rank identity →

This follows from the fact that the null spaces of

A

and

A^T A

are identical:

A\mathbf{x} = \mathbf{0}

implies

A^T A \mathbf{x} = \mathbf{0}

, and conversely

A^T A \mathbf{x} = \mathbf{0}

implies

\mathbf{x}^T A^T A \mathbf{x} = \|A\mathbf{x}\|^2 = 0

, so

A\mathbf{x} = \mathbf{0}

. By the rank-nullity theorem, equal nullities with the same

n

give equal ranks.

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Ranks readable from structure

For these types the rank follows from the defining condition, so no reduction is needed. Grouped by whether the structure pins the rank exactly or only bounds it.

5types

Full rank, guaranteed2

Identity§ 8

I_n

rank

n

whycolumns are the standard basis

nullity

0

The columns are independent by construction. Trivially the maximum, and the case every other row is measured against.

Symmetric positive definite§ 8

\mathbf{x}^{\mathsf{T}}A\mathbf{x} > 0

for

\mathbf{x} \neq \mathbf{0}

rank

n

whyno nonzero vector maps to zero

nullity

0

If

A\mathbf{x} = \mathbf{0}

then

\mathbf{x}^{\mathsf{T}}A\mathbf{x} = 0

, which the condition forbids unless

\mathbf{x} = \mathbf{0}

. So the null space is trivial and the rank is full — an argument that never mentions rows or columns.

Rank fixed by a count2

Diagonal§ 8

\operatorname{diag}(d_1, \ldots, d_n)

rankcount of

d_i \neq 0

whyeach nonzero entry is one pivot

nullitycount of

d_i = 0

Already in echelon form, so the pivots are the nonzero diagonal entries and no reduction happens. Rank and nullity are read off the same list.

Rank-one outer product§ 8

A = \mathbf{u}\mathbf{v}^{\mathsf{T}}

, both nonzero

rank

1

whyevery column is a multiple of

\mathbf{u}

nullity

n - 1

Column

j

is

v_j\mathbf{u}

, so the column space is the single line through

\mathbf{u}

however large the matrix is. These are the building blocks the SVD sums, and truncating that sum is low-rank approximation.

Only a bound1

Nilpotent§ 8

A^k = O

,

A \neq O

rank

< n

— strictly deficient

whya full-rank matrix has a full-rank power

nullity

\geq 1

The condition forces deficiency but does not say by how much: a

3 \times 3

nilpotent matrix may have rank

1

or

2

. This is the row where structure narrows the answer without determining it, and the reduction still has to be done.

What the reducible cases have in common is a definition that already says how many independent directions there are. A rank-one outer product has one because every column is a multiple of

\mathbf{u}

; a diagonal matrix has as many as it has nonzero entries. Where the definition is silent about independence — as with nilpotence — only a bound survives.

Ranks readable from structure·/linear-algebra/matrix/rankLearn Math Class

The last row is the instructive one because it does not behave like the others. Nilpotence forces the rank below

n

and stops there — a

3 \times 3

nilpotent matrix may have rank one or rank two, and nothing in the definition distinguishes them. Every other row here has a definition that already counts the independent directions; nilpotence only says that some direction collapses, which is a bound rather than a value.

Rank and the Four Fundamental Subspaces

Every

m \times n

matrix

A

gives rise to four subspaces, and the rank governs all of their dimensions.

The column space of

A

is the span of the columns, a subspace of

\mathbb{R}^m

with dimension equal to

\text{rank}(A)

. The row space of

A

is the span of the rows, a subspace of

\mathbb{R}^n

also with dimension

\text{rank}(A)

. The null space of

A

consists of all solutions to

A\mathbf{x} = \mathbf{0}

, a subspace of

\mathbb{R}^n

with dimension

n - \text{rank}(A)

. The left null space consists of all solutions to

A^T\mathbf{y} = \mathbf{0}

, a subspace of

\mathbb{R}^m

with dimension

m - \text{rank}(A)

.

These four subspaces split into two pairs of orthogonal complements. In

\mathbb{R}^n

, the row space and the null space are orthogonal complements: every vector in

\mathbb{R}^n

can be uniquely decomposed into a row-space component and a null-space component, and the two are perpendicular. In

\mathbb{R}^m

, the column space and the left null space form the analogous pair.

The four dimensions add up correctly on both sides:

\text{rank}(A) + (n - \text{rank}(A)) = n

in

\mathbb{R}^n

, and

\text{rank}(A) + (m - \text{rank}(A)) = m

in

\mathbb{R}^m

. The rank is the single number that controls the entire structural decomposition.

Subspace	Lives in	Description	Dimension
Column space of A	ℝᵐ	span of the columns of A	rank(A)
Row space of A	ℝⁿ	span of the rows of A	rank(A)
Null space of A	ℝⁿ	{x : Ax = 0}; orthogonal complement of the row space	n − rank(A)
Left null space of A	ℝᵐ	{y : Aᵀy = 0}; orthogonal complement of the column space	m − rank(A)

The reduced form all four spaces are read from

Pivot columns and free columns are both visible here, and between them they fix every one of the four spaces: the pivots give the dimensions of the row and column spaces, the free columns give the dimensions of the null spaces. One number, the rank, is doing all of that work at once. See each space extracted in turn on the four fundamental subspaces visualizer.

Rank is best thought of as the single number the four spaces are all sized against.

What Rank Governs

The rank shows up in nearly every structural fact about a matrix — solvability of systems, invertibility, the dimensions of the four fundamental subspaces, the image of the associated linear transformation. The table below collects these roles in one place: each row pairs an aspect of the matrix with what the single integer

r = \text{rank}(A)

pins down.

Aspect of A	What rank(A) = r determines	Reference
Effective dimensionality	r truly independent rows (= r independent columns); min(m, n) − r redundant directions	column rank = row rank
Column & row space dimension	dim(col(A)) = dim(row(A)) = r	four fundamental subspaces
Null space dimension	nullity(A) = n − r	rank-nullity theorem
Solvability of Ax = b	consistent iff rank(A) = rank([A \| b]); unique iff r = n; infinite iff r < n	three outcomes
Invertibility (square A)	A invertible iff r = n; equivalently det(A) ≠ 0	full rank condition
Linear transformation T(x) = Ax	dim(image(T)) = r; dim(kernel(T)) = n − r	dimension theorem

Matrix Rank FAQ

Does "full rank" mean the matrix is square?

+

No. Full rank means the rank has reached its ceiling of

\min(m, n)

, the shorter side of the matrix, so a

5 \times 3

matrix of rank

3

is at full rank while being nowhere near square. Falling short of that ceiling is called rank-deficient. The term measures attainment, not shape.Read more →

Why can you say "the rank" without specifying row or column rank?

+

Because a theorem licenses it. The columns live in

\mathbb{R}^m

and the rows in

\mathbb{R}^n

, genuinely different spaces, yet the two counts always coincide. Only that equality makes the unqualified word safe. It is a result rather than a definition, and something had to be proved before the bare term could exist.Read more →

What does $\rho(A)$ mean when you meet it?

+

It depends entirely on the source, so read the surrounding claim. Older and some engineering texts use

\rho(A)

for rank, but numerical analysis uses it for the spectral radius and statistics for correlation, and those are unrelated numbers. The unambiguous spellings are

\operatorname{rank}(A)

and its short form

\operatorname{rk}(A)

.Read more →

Rank of a Matrix

Measuring the Effective Size of a Matrix

What Rank Measures

Rank Notation

Rank Notation

Column Rank and Row Rank

Computing Rank via Row Reduction

Worked Example

Rank and Dimension

Rank and Linear Systems

The Rank-Nullity Theorem

Properties of Rank

Properties of rank

Rank of Special Matrices

Ranks readable from structure

Rank and the Four Fundamental Subspaces

What Rank Governs

Matrix Rank FAQ

Does "full rank" mean the matrix is square?

Why can you say "the rank" without specifying row or column rank?

What does $\rho(A)$ mean when you meet it?

Rank of a Matrix

Measuring the Effective Size of a Matrix

What Rank Measures

Rank Notation

Rank Notation

Column Rank and Row Rank

Computing Rank via Row Reduction

Worked Example

Rank and Dimension

Rank and Linear Systems

The Rank-Nullity Theorem

Properties of Rank

Properties of rank

Rank of Special Matrices

Ranks readable from structure

Rank and the Four Fundamental Subspaces

What Rank Governs

Matrix Rank FAQ

Does "full rank" mean the matrix is square?

Why can you say "the rank" without specifying row or column rank?

What does ρ(A)\rho(A)ρ(A) mean when you meet it?

What does $\rho(A)$ mean when you meet it?