For a nonsingular square matrix A, its condition number is Cond(A)=∥A∥∥∥A−1∥∥=(x=0max∥x∥∥Ax∥)(x=0min∥x∥∥Ax∥)−1
The condition number of a matrix denotes the ratio of the maximum relative stretching to the maximum relative shrinking to any nonzero vectors. In other words, it measures the amount of distortion of the unit sphere. Therefore, the condition number is not large when A is a scaling matrix which stretches evenly to all the axes. When A stretches espacially more to some axes, the condition number is larger. By convention, Cond(A)=∞ if A is singular.
Euclidean condition number is
where λmax(X) and λmin(X) are the maximum and minimum eigenvalues of X. If A symmetric, then AtAx=A(λx)=λ2x where x and λ are the eigenvector and eigenvalue of A, so ∥A∥2∥∥A−1∥∥2=λmin(AtA)λmax(AtA)=∣∣λmin(A)λmax(A)∣∣
The large the condition number is, the large the amount of distortion of the unit sphere is.
Cond(A)≥1 and Cond(I)=1.
Cond(γA)=Cond(A) where γ∈R.
Cond(D)=min∣di∣max∣di∣ where D is a diagonal matrix and di is the i-th element of D.
The condition number A tells us about how close to singularity A is. On the other hand, detA is not a good measure. When A is singular, detA=0, but it does not mean that A is close to singularity. For example, detαIn=αn for α∈R. For ∣α∣<1, however, the determinant is close to zero although In is perfectly well-conditioned.
When estimating the condition number, the hardest part would be calculating ∥∥A−1∥∥. Suppose that Az=y and A is nonsingular. Then ∥z∥=∥∥A−1y∥∥≤∥∥A−1∥∥∥y∥, so ∥y∥∥z∥≤∥∥A−1∥∥. If a good y is selected which makes ∥y∥∥z∥ large enough, it can approximate to ∥∥A−1∥∥. First of all, it can be found by random. Second, it can be solved by Aty=c where c is a vector whose elements are 1 or −1 so that c makes y as large as possible. Finally, it can be ∣∣λmin(A)λmax(A)∣∣ in case of estimating Euclidean condition number.
 Michael T. Heath, Scientific Computing: An Introductory Survey. 2nd Edition, McGraw-Hill Higher Education.
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