Rank of two-way arrays

Rank is a very important mathematical property of two-way arrays (matrices). The mathematical definition of rank is unambiguous, but for chemistry this rank definition is not always the most convenient one. In the first section, different rank definitions are given with examples. The rank of three-way and multi-way arrays is considerably more complicated than for the two-way arrays and is treated separately. 

Suppose that a matrix A (7 x /) is given. The matrix A has J columns. The number of linearly independent columns of A is called the column-rank of A.1 The matrix A defines a linear transformation from R7 to R7. The range or column space of A is defined as 31(A) = y Ax = y Vx e R7 or, stated in words, the range of A consists of all the ys for which there is an x such that Ax = y. Clearly, 31(A) c R1. The rank of A is the dimension of 31(A) and this equals the column-rank of A. 

The idea of rank and column-rank can be illustrated with an example. Suppose that 

Similarly, the number of linearly independent rows of A is called the row-rank of A. The row-rank of A is the column-rank of A. A fundamental theorem in matrix algebra states that the row-rank and the column-rank of a matrix are equal (and equal to the rank) [Schott 1997], Hence, it follows that the rank r(A) min(/,/). The matrix A has full rank if and only if r(A) = mini/,/). Sometimes the term full column-rank is used. This means that r(A) = min(/,/) =. /, implying that J I. The term full row-rank is defined analogously. 

Some important equalities follow. Suppose A is (I x J) and the rank r(A) = R, then  

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