diff --git a/src/index.md b/src/index.md index 6fa21e2bf..9e7b913a3 100644 --- a/src/index.md +++ b/src/index.md @@ -63,6 +63,7 @@ especially popular in field of competitive programming.* - [Gauss & System of Linear Equations](./linear_algebra/linear-system-gauss.html) - [Gauss & Determinant](./linear_algebra/determinant-gauss.html) - [Kraut & Determinant](./linear_algebra/determinant-kraut.html) +- [Rank of a matrix](./linear_algebra/rank-matrix.html) ### Combinatorics - [The Inclusion-Exclusion Principle](./combinatorics/inclusion-exclusion.html) diff --git a/src/linear_algebra/rank-matrix.md b/src/linear_algebra/rank-matrix.md new file mode 100644 index 000000000..6a7050a35 --- /dev/null +++ b/src/linear_algebra/rank-matrix.md @@ -0,0 +1,55 @@ + + +#Finding the rank of a matrix + +**The rank of a matrix** is the largest number of linearly independent rows/columns of the matrix. The rank is not only defined for square matrices. + +The rank of a matrix can also be defined as the largest order of any non-zero minor in the matrix. + +Let the matrix be rectangular and have size $N \times M$. +Note that if the matrix is square and its determinant is non-zero, then the rank is $N$ ($=M$); otherwise it will be less. Generally, the rank of a matrix does not exceed $\min (N, M)$. + +##Algorithm + +You can search for the rank using [Gaussian elimination](./linear_algebra/linear-system-gauss.html). We will perform the same operations as when solving the system or finding its determinant. But if at any step in the $i$-th column there are no rows with an non-empty entry among those that we didn't selected already, then we skip this step and decrease the rank by one (initially the rank is set equal to $\max (N, M)$). +Otherwise, if we have found a row with a non-zero element in the $i$-th column during the $i$-th step, then we mark this row as a selected one and perform the usual operations of taking this row away from the rest. + +##Complexity + +This algorithm runs in $\mathcal{O}(n^3)$. + +##Implementation + +```cpp +const double EPS = 1E-9; + +int compute_rank(vector> A) { + int n = A.size(); + int m = A[0].size(); + + int rank = max(n, m); + vector row_selected(n, false); + for (int i = 0; i < m; ++i) { + int j; + for (j = 0; j < n; ++j) { + if (!row_selected[j] && abs(A[j][i]) > EPS) + break; + } + + if (j == n) { + --rank; + } else { + row_selected[j] = true; + for (int p = i + 1; p < m; ++p) + A[j][p] /= A[j][i]; + for (int k = 0; k < n; ++k) { + if (k != j && abs(A[k][i]) > EPS) { + for (int p = i + 1; p < m; ++p) + A[k][p] -= A[j][p] * A[k][i]; + } + } + } + } + return rank; +} +``` pFad - Phonifier reborn

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