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The rank of A is the smallest number k such that A can be written as a sum of k rank 1 matrices, where a matrix is defined to have rank 1 if and only if it can be written as a nonzero product of a column vector c and a row vector r.
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 rank of a matrix A is the maximum number of linearly independent row ... Calculating the matrix product of two n-by-n matrices using the definition given above ...
Matrix multiplication is thus a basic tool of linear algebra, and as such has numerous applications in many areas of mathematics, as well as in applied mathematics, statistics, physics, economics, and engineering. [3] [4] Computing matrix products is a central operation in all computational applications of linear algebra.
The rank of a matrix is the minimum number of column vectors needed to span the range of the matrix. A matrix thus has rank one if it can be written as an outer product of two nonzero vectors: =. The rank of a matrix A is the smallest number of such outer products that can be summed to produce it: = + +.
Apart from the definition given above, there is the following: The nonnegative rank of a nonnegative m×n-matrix A is equal to the smallest number q such there exists a nonnegative m×q-matrix B and a nonnegative q×n-matrix C such that A = BC (the usual matrix product). To obtain the linear rank, drop the condition that B and C must be ...
Suppose a vector norm ‖ ‖ on and a vector norm ‖ ‖ on are given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows: ‖ ‖, = {‖ ‖: ‖ ‖ =} = {‖ ‖ ‖ ‖:} . where denotes the supremum.
The rank of the matrix is therefore even, and is ... one or the other definition of the product could be used, as the two algebras are isomorphic ...