An array is defined to have a shape given by its number of dimensions (called "rank") and the extent for each one of these. Two arrays agree if they have the same shape. Operations are normally done element for element. Please note that the rank of an array is the number of dimensions and has nothing to do with the mathematical rank of a matrix!
In the following simple example I show how you can assign matrices with simple statements like B = A, how you can use the intrinsic matrix multiplication MATMUL and the addition SUM and how you can use the array sections (in the example below I use array sections who are vectors).
PROGRAM ARRAY_EXAMPLE IMPLICIT NONE INTEGER :: I, J REAL, DIMENSION (4,4) :: A, B, C, D, E DO I = 1, 4 ! calculate a test matrix DO J = 1, 4 A(I, J) = (I-1.2)**J END DO END DO B = A*A ! element for element multiplication CALL PRINTF(A,4) ; CALL PRINTF(B,4) C = MATMUL(A, B) ! internal matrix multiplication DO I = 1, 4 ! explicit matrix multiplication DO J = 1, 4 D(I, J) = SUM( A(I,:)*B(:,J) ) END DO END DO CALL PRINTF(C,4) ; CALL PRINTF(D,4) E = C - D ! comparison of the two methods CALL PRINTF(E,4) CONTAINS SUBROUTINE PRINTF(A, N) ! print an array IMPLICIT NONE INTEGER :: N, I REAL, DIMENSION (N, N) :: A DO I = 1, N WRITE(*,' (4E15.6)') A(I,:) END DO WRITE(*,*) ! write the blank line END SUBROUTINE PRINTF END PROGRAM ARRAY_EXAMPLEAs was mentioned in chapter 9 about recursion, functions in Fortran 90 can be array valued. In that case it is recommended to use the RESULT property to specify a result variable that is supposed to store the array.
Fortran 90 has much larger possibilities than Fortran 77 to permit dynamic memory allocation, which in Fortran 77 only could be done when a sufficient storage area had been allocated in the calling program unit, and both the array name and the required dimension(s) have to be included as parameters in the call of the subprogram. This is the concept adjustable array. A very simple case is the one with the last dimension, which can be given simply with a *, assumed-size array.
Now we also have the possibilities allocatable array, automatic array, and assumed-shape array. Dynamic allocation using pointers is discussed in a section of the next chapter. An overview is given in Appendix 3 (section 10), and see also Appendix 9 (explanation of certain terms).