Vivette Girault is not a widely recognized figure or a concept that has well-documented information available in popular sources. It's possible that Vivette Girault could refer to a specific individual, possibly in a niche field or a lesser-known context.

A block matrix is a matrix that is partitioned into smaller matrices, known as "blocks." These smaller matrices can be of different sizes and can be arranged in a rectangular grid format. Block matrices are particularly useful in various mathematical fields, including linear algebra, numerical analysis, and optimization, as they allow for simpler manipulation and operations on large matrices. ### Structure of Block Matrices A matrix \( A \) can be represented as a block matrix if it is partitioned into submatrices.

The RV Wecoma was a research vessel operated by OSU (Oregon State University), primarily used for marine science and oceanographic research. Launched in 1996 and taking its name from "Wecoma" (a portmanteau of "West Coast" and "Oregon"), the RV Wecoma had a versatile design suitable for a variety of research activities, including oceanographic studies, fisheries research, and marine biology.

A Butson-type Hadamard matrix is a generalization of Hadamard matrices that is defined for complex entries and is characterized by its entries being roots of unity.

The cross-covariance matrix is a statistical tool that captures the covariance between two different random vectors (or random variables). Specifically, it quantifies how much two random variables change together. Unlike the covariance matrix, which involves the variances of a single random vector, the cross-covariance matrix deals with the relationships between different vectors.

The Cross Gramian is a mathematical construct used in the fields of control theory, signal processing, and systems theory. It is primarily associated with the analysis of linear time-invariant (LTI) systems and helps in understanding the relationships between different input-output systems. Given two linear systems described by their state-space representations, the Cross Gramian can be used to quantify the interaction between these systems. Specifically, it can be applied to determine controllability and observability properties when dealing with multiple systems.

In linear algebra, a definite matrix refers to a square matrix that has specific properties related to the positivity of its quadratic forms. The terminology typically includes several definitions: 1. **Positive Definite Matrix**: A symmetric matrix \( A \) is called positive definite if for all non-zero vectors \( x \), the following holds: \[ x^T A x > 0. \] This implies that all eigenvalues of the matrix are positive.

In the context of solving linear differential equations, a **fundamental matrix** refers to a matrix that plays a critical role in finding the general solution to a system of first-order linear differential equations.

A generalized permutation matrix is a broader concept than a standard permutation matrix, which is a square matrix used to permute the elements of vectors in linear algebra. While a standard permutation matrix contains exactly one entry of 1 in each row and each column, with all other entries being 0, a generalized permutation matrix allows for more flexibility.

A generalized inverse of a matrix is a broader concept than the ordinary matrix inverse, which only exists for square matrices that are nonsingular (i.e., matrices that have a non-zero determinant). Generalized inverses can be defined for any matrix, whether it is square, rectangular, singular, or nonsingular. ### Types of Generalized Inverses The most commonly used type of generalized inverse is the Moore-Penrose pseudoinverse.

The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function. It provides important information about the local curvature of the function and is widely used in optimization problems, economics, and many areas of mathematics and engineering.

Hierarchical matrices, often referred to as H-matrices, are a data structure and mathematical framework used to efficiently represent and compute with large, sparse matrices, particularly those that arise in applications related to numerical analysis, scientific computing, and simulations. The main idea behind H-matrices is to exploit the hierarchical structure of the matrix by grouping data in a way that captures its sparsity while enabling efficient operations like matrix-vector multiplication and matrix-matrix multiplication.

The Householder transformation is a linear algebra technique used to perform orthogonal transformations of vectors and matrices. It is particularly useful in numerical linear algebra for QR decomposition and in other applications where one needs to reflect a vector across a hyperplane defined by another vector.

An involutory matrix is a square matrix \( A \) that satisfies the property: \[ A^2 = I \] where \( I \) is the identity matrix of the same dimension as \( A \). This means that when the matrix is multiplied by itself, the result is the identity matrix.

The Moore determinant, also known as the Moore-Penrose determinant, is a generalization of the determinant for matrices that may not be square or may not have full rank. However, it primarily caters to the needs of generalized inverses in the context of singular matrices.

A Manin matrix, named after the mathematician Yuri I. Manin, is a specific type of matrix that arises in various mathematical contexts, particularly in relation to the study of linear systems, algebraic geometry, and representation theory. In a more precise mathematical context, a Manin matrix is often discussed in the framework of certain algebraic structures (such as algebraic groups or varieties) where it can exhibit particular properties related to linearity, symmetries, or transformations.

A Vandermonde matrix is a specific type of matrix with a particular structure, commonly used in polynomial interpolation and linear algebra.

The Nekrasov matrix is a concept that arises in the context of mathematical physics, particularly in the study of supersymmetric gauge theories and their connections to algebraic geometry and integrable systems. It is named after the Russian mathematician Nikita Nekrasov, who contributed significantly to the field.

An orthostochastic matrix is a mathematical construct that arises in the context of stochastic processes and linear algebra. Specifically, it is a type of matrix associated with stochastic transformations, preserving certain probabilistic properties. A matrix \( A \) is termed orthostochastic if it satisfies the following conditions: 1. **Non-negativity:** All entries of \( A \) are non-negative, meaning \( a_{ij} \geq 0 \) for all entries \( i, j \).

The UK Molecular R-matrix Codes are a set of computational tools used for performing quantum mechanical calculations in atomic and molecular physics, particularly in the context of scattering and photoionization processes. The R-matrix method itself is a highly versatile and powerful approach used to solve the Schrödinger equation for multi-electron systems in various interaction scenarios.