Search results
Results from the WOW.Com Content Network
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.
A matrix that has rank min(m, n) is said to have full rank; otherwise, the matrix is rank deficient. Only a zero matrix has rank zero. f is injective (or "one-to-one") if and only if A has rank n (in this case, we say that A has full column rank). f is surjective (or "onto") if and only if A has rank m (in this case, we say that A has full row ...
Every finite-dimensional matrix has a rank decomposition: Let be an matrix whose column rank is . Therefore, there are r {\textstyle r} linearly independent columns in A {\textstyle A} ; equivalently, the dimension of the column space of A {\textstyle A} is r {\textstyle r} .
An alternate way of summarizing the design trials would be to use a 4x3 matrix whose 4 rows are the levels of the treatment X 1 and whose columns are the 3 levels of the blocking variable X 2. The cells in the matrix have indices that match the X 1, X 2 combinations above.
Each column containing a leading 1 has zeros in all entries above the leading 1. While a matrix may have several echelon forms, its reduced echelon form is unique. Given a matrix in reduced row echelon form, if one permutes the columns in order to have the leading 1 of the i th row in the i th column, one gets a matrix of the form
where R 1 is an n×n upper triangular matrix, 0 is an (m − n)×n zero matrix, Q 1 is m×n, Q 2 is m×(m − n), and Q 1 and Q 2 both have orthogonal columns. Golub & Van Loan (1996 , §5.2) call Q 1 R 1 the thin QR factorization of A ; Trefethen and Bau call this the reduced QR factorization . [ 1 ]
For the case of column vector c and row vector r, each with m components, the formula allows quick calculation of the determinant of a matrix that differs from the identity matrix by a matrix of rank 1: (+) = +. More generally, [14] for any invertible m × m matrix X,
Applicable to: m-by-n matrix A of rank r Decomposition: A = C F {\displaystyle A=CF} where C is an m -by- r full column rank matrix and F is an r -by- n full row rank matrix Comment: The rank factorization can be used to compute the Moore–Penrose pseudoinverse of A , [ 2 ] which one can apply to obtain all solutions of the linear system A x ...