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A matrix has rank at least exactly when it has a nonzero -rowed minor
Statement
Let and let . Then if and only if some -rowed minor of is nonzero (Submatrices and minors of a rectangular matrix). Equivalently, the rank of is the largest positive size of a nonzero minor when , and it is when every entry is zero (The rank of a matrix equals the rank of the linear map ).
Facts & Assumptions
Given: A real matrix and a natural with .
The rank of a matrix is the dimension of its column space, because row rank equals column rank (Row space, column space, nullspace, row rank, column rank and matrix rank, Row rank equals column rank, and both equal the number of pivots); a list is independent exactly when only the zero coefficient list has zero linear combination (Linear independence: a finite list is independent when forces every , and a subset is independent when every injective finite list into is independent).
A real matrix has rank exactly when it is invertible, and it is invertible exactly when its determinant is nonzero (Invertible matrix theorem: invertibility, full pivot rank, RREF , trivial nullspace and unique solvability are equivalent, A finite square real matrix is invertible if and only if its determinant is nonzero).
Proof
Suppose first that the -minor is nonzero, and write for the corresponding submatrix.
Conversely, suppose . Choose independent columns and form the resulting matrix . Its column rank and hence its row rank are , so its rows span and contain independent rows.
By [L2], is invertible. If a linear combination of the columns of indexed by vanishes, restricting that equality to the rows in gives , hence . Those columns are independent, so .
The submatrix determined by those columns and rows has rank , so [L2] makes its determinant nonzero. This is an -rowed minor of , proving the converse and the equivalence.
Depends on
- Submatrices and minors of a rectangular matrix
- Row space, column space, nullspace, row rank, column rank and matrix rank
- Row rank equals column rank, and both equal the number of pivots
- The rank of a matrix equals the rank of the linear map $x\mapsto Ax$
- Invertible matrix theorem: invertibility, full pivot rank, RREF $I$, trivial nullspace and unique solvability are equivalent
- A finite square real matrix is invertible if and only if its determinant is nonzero
- Linear independence: a finite list $v : n \to V$ is independent when $\sum_{i<n} \lambda_i v_i = 0_V$ forces every $\lambda_i = 0_F$, and a subset $S \subseteq V$ is independent when every injective finite list into $S$ is independent
Used by
- The map (x,y)↦(x,xy) has nonconstant rank on every neighbourhood of the origin Example
- Two constraints on a sphere-plane circle, where one multiplier solution is only a local maximum Example
- A nonzero rank minor supplies the source coordinates for the constant-rank theorem Lemma
- Deleted-row minors of a zero-column-sum matrix agree up to sign Lemma
- In source rank coordinates, the remaining components depend only on the rank coordinates Lemma
- Weak containment of the trivial representation and almost invariant vectors Lemma
- Differential rank is lower semicontinuous Theorem
Dependency tree · two levels
39 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. M. Lee, Introduction to Smooth Manifolds, Theorem 7.13 (standard reference, not scraped)
- L. W. Tu, An Introduction to Manifolds, Section 11.1 (standard reference, not scraped)