Rank, orthogonality, and column span in a 3x3 matrix

The identity matrix has a two-norm condition number of one. This is because its columns are orthonormal, meaning they are mutually perpendicular and have unit length, which represents the most stable configuration for linear calculations.

Conditions: The matrix is the identity matrix.; The condition number is measured in the two-norm.

A column space can have lower dimension than the ambient coordinate space when the columns of the matrix are linearly dependent. In the example, the columns span the xy-plane, which is a two-dimensional subspace of the three-dimensional ambient space, resulting in a matrix rank of two.

Conditions: The matrix has three columns in a three-dimensional ambient space.; The columns are linearly dependent and confined to a plane.

Giving a column a nonzero component in a previously unused dimension makes the columns linearly independent, thereby increasing the exact rank to full rank. Even if this perturbation is tiny, the exact mathematical rank changes, although the columns remain nearly dependent.

Conditions: The original columns are linearly dependent.; A perturbation adds a nonzero component to a column in a new dimension.

Near-dependence of columns can produce a much larger condition number. While the exact rank might be full, the geometric closeness of the columns to a lower-dimensional subspace makes the matrix ill-conditioned, leading to sensitivity in calculations.

Conditions: The columns are linearly independent but nearly dependent.; The matrix has full exact rank.

General orthogonal matrices require columns that are both unit-length and mutually perpendicular. These two properties together define orthonormal columns, which ensure optimal numerical conditioning.

Conditions: The matrix is classified as an orthogonal matrix.