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A matrix is said to have full rank if its rank equals the largest possible for a matrix of the same dimensions, which is the lesser of the number of rows and columns. A matrix is said to be rank-deficient if it does not have full rank. The rank deficiency of a matrix is the difference between the lesser of the number of rows and columns, and ...
Even though the row is indicated by the first index and the column by the second index, no grouping order between the dimensions is implied by this. The choice of how to group and order the indices, either by row-major or column-major methods, is thus a matter of convention. The same terminology can be applied to even higher dimensional arrays.
The dimension of the row space is called the rank of the matrix. This is the same as the maximum number of linearly independent rows that can be chosen from the matrix, or equivalently the number of pivots. For example, the 3 × 3 matrix in the example above has rank two. [9] The rank of a matrix is also equal to the dimension of the column space.
The solution is unique if and only if the rank equals the number of variables . Otherwise the general solution has free parameters where is the difference between the number of variables and the rank. In such a case there as an affine space of solutions of dimension equal to this difference.
The rank of a matrix is equal to the dimension of the row space, so row equivalent matrices must have the same rank. This is equal to the number of pivots in the reduced row echelon form. A matrix is invertible if and only if it is row equivalent to the identity matrix.
Predating the term rank (sometimes also called row) is the use of single-sided and double-sided modules, especially with SIMMs. While most often the number of sides used to carry RAM chips corresponded to the number of ranks, sometimes they did not. This could lead to confusion and technical issues. [2] [3]
The number of indices needed to specify an element is called the dimension, dimensionality, or rank of the array. In standard arrays, each index is restricted to a certain range of consecutive integers (or consecutive values of some enumerated type ), and the address of an element is computed by a "linear" formula on the indices.
For the cases where has full row or column rank, and the inverse of the correlation matrix ( for with full row rank or for full column rank) is already known, the pseudoinverse for matrices related to can be computed by applying the Sherman–Morrison–Woodbury formula to update the inverse of the ...