Let A be a nonsingular square matrix. Prove the following...

Let $A$ be a nonsingular square matrix. Prove the following formulas for its condition number:
(a) $\kappa(A)=\frac{\max \{\|A \mathbf{u}\| \mid\|\mathbf{u}\|=1\}}{\min \{\|A \mathbf{u}\| \mid\|\mathbf{u}\|=1\}}$,
(b) $\kappa(A)=\|A\|_2\left\|A^{-1}\right\|_2$.

Key Concepts

Inverse of a Matrix

The inverse of a nonsingular matrix reverses the effect of the matrix. In the context of the condition number, understanding the inverse is crucial because the size of A's inverse (measured in a compatible norm) directly affects the sensitivity of the system. The relation between the norms of a matrix and its inverse underpins many bounds and estimates used in numerical linear algebra.

Operator Norm (Induced Norm)

An operator norm, also called an induced norm, is defined by taking the maximum of the norm of the output vector A u when the norm of the input vector u is fixed to one. It provides a measure of the maximum amplification factor of the matrix and plays a key role in expressing the condition number as the ratio of the maximum to minimum values obtained on the unit sphere.

Singular Values

Singular values of a matrix are the square roots of the eigenvalues of the matrix A*A (or AA*). They offer insights into the geometric action of the matrix, including stretching factors along principal directions. The largest singular value corresponds to the operator norm, while the smallest singular value, when the matrix is nonsingular, is inversely related to the norm of the inverse and together these encapsulate the condition number.

Matrix Norm

A matrix norm is a function that assigns a non-negative scalar to a matrix, typically reflecting the size or length of the matrix in a certain sense. It is often defined in a way that it is consistent with a vector norm, known as an induced or operator norm, which measures how much the matrix can 'stretch' a vector.

Condition Number

The condition number of a matrix quantifies how the output value of a function or system is affected by small changes in the input. In the context of linear algebra, it measures the sensitivity of the solution of a system of linear equations to errors or perturbations in the input data, and is a fundamental property in numerical analysis for assessing stability and accuracy of computations.

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