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.
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.
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.
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.