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Rank–nullity theorem. The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity of M; and; the dimension of the domain of a linear transformation f is the sum of the rank of f (the dimension of the image of f) and the nullity of f (the dimension of ...
The nullity of M is given by m − n + c, where, c is the number of components of the graph and n − c is the rank of the oriented incidence matrix. This name is rarely used; the number is more commonly known as the cycle rank, cyclomatic number, or circuit rank of the graph. It is equal to the rank of the cographic matroid of the graph.
The nullity of a matrix is the dimension of the null space, and is equal to the number of columns in the reduced row echelon form that do not have pivots. [7] The rank and nullity of a matrix A with n columns are related by the equation:
Now, each row of A is given by a linear combination of the r rows of R. Therefore, the rows of R form a spanning set of the row space of A and, by the Steinitz exchange lemma, the row rank of A cannot exceed r. This proves that the row rank of A is less than or equal to the column rank of A.
Analogously, the nullity of the graph is the nullity of its oriented incidence matrix, given by the formula m − n + c, where n and c are as above and m is the number of edges in the graph. The nullity is equal to the first Betti number of the graph. The sum of the rank and the nullity is the number of edges.
Then the rank function of M is: r(B) = min(k, |B|). Let M be a partition matroid: the elements of E are partitioned into categories, each category c has capacity k c, and the independent sets are those containing at most k c elements of category c. Then the rank function of M is: r(B) = sum c min(k c, |B c |) where B c is the subset B contained ...
By left-multiplication with an appropriate invertible matrix L, it can be achieved that row t of the matrix product is the sum of σ times the original row t and τ times the original row k, that row k of the product is another linear combination of those original rows, and that all other rows are unchanged.
The nullity, N, of a graph with s separate parts and b branches is defined by: = + The nullity of a graph represents the number of degrees of freedom of its set of network equations. For a planar graph, the nullity is equal to the number of meshes in the graph. [34] The rank, R of a graph is defined by: