Math 2.0

Linear and multilinear algebra; matrix theory Systems of linear equations Proposition

Rank of a product of matrices

Builds on Rank of a matrixMatrix multiplication

Statement

Let FF be a field, let AA be an m×nm \times n matrix and BB an n×pn \times p matrix over FF. Then rank⁡(AB)≤min⁡(rank⁡A,rank⁡B),\rank(AB) \le \min(\rank A, \rank B), where rank⁡\rank is the rank of a matrix.

Remarks

The column space of ABAB is contained in the column space of AA, so rank⁡(AB)≤rank⁡A\rank(AB) \le \rank A. The row space of ABAB is contained in the row space of BB, so rank⁡(AB)≤rank⁡B\rank(AB) \le \rank B. If AA is invertible, then rank⁡(AB)=rank⁡B\rank(AB) = \rank B; if BB is invertible, then rank⁡(AB)=rank⁡A\rank(AB) = \rank A. A lower bound is Sylvester's rank inequality: rank⁡A+rank⁡B−n≤rank⁡(AB)\rank A + \rank B - n \le \rank(AB).

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Matrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionMatrix multiplication, definition, no evidence levelMatrix multiplicationdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionColumn space, definition, no evidence levelColumn spacedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionDimension, definition, no evidence levelDimensiondefinitionRank of a matrix, definition, no evidence levelRank of a matrixdefinitionRank of a product of matrices, statement, no evidence levelRank of a product of …statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Matrix definition
  2. Vector space definition
  3. Matrix multiplication definition
  4. Linear combination definition
  5. Linear subspace definition
  6. Linear independence definition
  7. Linear span definition
  8. Column space definition
  9. Spanning set definition
  10. Basis definition
  11. Finite-dimensional vector space definition
  12. Dimension definition
  13. Rank of a matrix definition
  14. Rank of a product of matrices statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Rank_(linear_algebra), Properties
About this page
Tag02FDWY55AS
URIm2://02FDWY55AS
Kindstatement, proposition
WritingW0 stub
MSC15A03
StatusThis copy of the site has no build data. Run m2 status 02FDWY55AS.
UsesRank of a matrix, Matrix multiplication

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