## What Is the Gerstenhaber Problem?

When Cayley introduced matrix algebra in 1858, he did much more than merely arrange numbers in a rectangular array. His definitions of addition, multiplication, and inversion produced an algebraic structure that has proved to be immensely useful, and which still … Continue reading

Posted in what-is | 4 Comments

## What Is a Cholesky Factorization?

The Cholesky factorization of a symmetric positive definite matrix is the factorization , where is upper triangular with positive diagonal elements. It is a generalization of the property that a positive real number has a unique positive square root. The … Continue reading

## What is Numerical Stability?

Numerical stability concerns how errors introduced during the execution of an algorithm affect the result. It is a property of an algorithm rather than the problem being solved. I will assume that the errors under consideration are rounding errors, but … Continue reading

## What is the Polar Decomposition?

A polar decomposition of with is a factorization , where has orthonormal columns and is Hermitian positive semidefinite. This decomposition is a generalization of the polar representation of a complex number, where corresponds to and to . When is real, … Continue reading

## What Is a Symmetric Positive Definite Matrix?

A real matrix is symmetric positive definite if it is symmetric ( is equal to its transpose, ) and By making particular choices of in this definition we can derive the inequalities Satisfying these inequalities is not sufficient for positive … Continue reading

## What Is the Growth Factor for Gaussian Elimination?

Gaussian elimination is the process of reducing an matrix to upper triangular form by elementary row operations. It consists of stages, in the th of which multiples of row are added to later rows to eliminate elements below the diagonal … Continue reading

## What Is Stochastic Rounding?

In finite precision arithmetic the result of an elementary arithmetic operation does not generally lie in the underlying number system, , so it must be mapped back into by the process called rounding. The most common choice is round to … Continue reading

## What Is the Hilbert Matrix?

The Hilbert matrix is the matrix with . For example, It is probably the most famous test matrix and its conditioning and other properties were extensively studied in the early days of digital computing, especially by John Todd. The Hilbert … Continue reading

Posted in what-is | 3 Comments

## What Is a Fréchet Derivative?

Let and be Banach spaces (complete normed vector spaces). The Fréchet derivative of a function at is a linear mapping such that for all . The notation should be read as “the Fréchet derivative of at in the direction ”. … Continue reading