Numerical Methods with Programming

Section 1.5 Accessing the entries of vectors and matrices

We often need to manipulate matrices and vectors by extracting, replacing or rearranging some elements. The elements are indexed starting with 1: so, after executing v = [8 6 -4] we find that v(1) is 8, v(2) is 6, and v(3) is -4. One can also use indexing from the end of the vector: v(end) is -4, v(end-1) is 6, and so on. For a matrix the indexes are ordered as (row, column). So, if A = [5 6 7; 9 8 6], then A(1, 3) is 7 and A(2, 1) is 9.

A powerful tool for working with matrices is the colon : selector. When it replaces an index, it means “run through all values of that index”. Given a matrix A, we can use A(1, :) to get its first row, A(:, 2) to get the second column, and A(:, end) for the last column.

We can use a vector of indexes to extract several elements of a vector or a matrix. For example, v(3:end-1) extracts all entries of v starting with the 3rd one and ending with the one before the last. If v was [8 6 4 3 2 1], the result would be [4 3 2]. For another example, v(2:2:end) extracts all even-numbered elements of v. In the above example this would be [6 3 1]. For a matrix A, we can select both rows and columns at the same time: A(2:3, 2:5) means taking the elements of A that appear in rows 2-3 and in columns 2-5. The result is a submatrix of size 2×4.

The entries can also be assigned, for example v(2:2:end) = -3 makes all even-numbered entries of vector v equal to -3.

A = [1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16];
B = A(2:3, 2:3);
disp(B)