The condition number in matrix computations measures what aspect?

The condition number in matrix computations is a critical concept that measures the sensitivity of the output of a matrix operation to small changes in the input. Specifically, it quantifies how much the solution of a linear system can change in response to small perturbations in the input data or the coefficients of the matrix.

In practical terms, a high condition number indicates that the matrix is ill-conditioned, meaning that even tiny errors in input can lead to significant fluctuations in the output. Conversely, a low condition number suggests that the matrix is well-conditioned, and small errors in input will result in proportionately small changes in the output. This sensitivity analysis is vital for understanding the reliability of numerical solutions in engineering and computational applications.

The other aspects mentioned do not pertain to the condition number. The size of the matrix refers to its dimensions, the number of non-zero elements relates to the matrix's sparsity, and symmetry concerns the arrangement of elements rather than the sensitivity of the system’s solutions. Each of these factors plays a role in different areas of matrix theory but does not address the concept of condition numbers directly.