Applied mathematics, numerical linear algebra and software.
The spectral radius of a square matrix is the largest absolute value of any eigenvalue of : For Hermitian matrices (or more generally normal matrices, those satisfying ) the spectral radius is just the -norm, . What follows is most interesting for nonnormal matrices. Two classes of matrices for which the spectral radius is known Continue reading What Is the Spectral Radius of a Matrix?
An upper Hessenberg matrix has the property that for . For , the structure is is an upper triangular matrix with an extra subdiagonal. A lower Hessenberg matrix is the transpose of an upper Hessenberg matrix. In the rest of this article, the Hessenberg matrices are upper Hessenberg. Hessenberg matrices play a key role in Continue reading What Is a Hessenberg Matrix?
An upper bidiagonal matrix depends on just parameters, which appear on the main diagonal and the superdiagonal. Such matrices arise commonly, for example as the factor and the transpose of the factor in the LU factorization of tridiagonal matrices, and as the intermediate matrix in the computation of the singular value decomposition by the Golub–Reinsch Continue reading The Power of…
A subspace of is an invariant subspace for if , that is, if implies . Here are some examples of invariant subspaces. and are trivially invariant subspaces of any . The null space is an invariant subspace of because implies . If is an eigenvector of then is a -dimensional invariant subspace, since , where Continue reading What Is an Invariant Subspace?
A submatrix of a matrix is another matrix obtained by forming the intersection of certain rows and columns, or equivalently by deleting certain rows and columns. More precisely, let be an matrix and let and . Then the matrix with is the submatrix of comprising the elements at the intersection of the rows indexed by Continue reading What Is a Submatrix?
A flop is one of the elementary arithmetic operations , , , carried out on floating-point numbers. For example, evaluating the expression takes three flops. A square root, which occurs infrequently in numerical computation, is also counted as one flop. As an example, the computation of the inner product of two -vectors and can be Continue reading What Is a Flop?
Complex numbers have the form where is the imaginary unit. Quaternions contain two more imaginary units, and : Sir William Rowan Hamilton discovered the quaternions in 1843 as he walked along the Royal Canal in Dublin, and he famously carved the formulas on a stone of Brougham Bridge. He had for some time been trying Continue reading Hamilton, the Quaternions, and Creativity
The pseudoinverse is an extension of the concept of the inverse of a nonsingular square matrix to singular matrices and rectangular matrices. It is one of many generalized inverses, but the one most useful in practice as it has a number of special properties. The pseudoinverse of a matrix is an matrix that satisfies the Continue reading What Is the Pseudoinverse of a Matrix?
The numerical range of a matrix , also known as the field of values, is the set of complex numbers The set is compact and convex (a nontrivial property proved by Toeplitz and Hausdorff), and it contains all the eigenvalues of . For normal matrices it is the convex hull of the eigenvalues. For a Continue reading What Is the Numerical Range of a Matrix?
James Wilkinson s 1963 book Rounding Errors in Algebraic Processes has been hugely influential. It came at a time when the effects of rounding errors on numerical computations in finite precision arithmetic had just starting to be understood, largely due to Wilkinson s pioneering work over the previous two decades. The book gives a uniform treatment of Continue reading Wilkinson s Rounding Errors…