The Quadratic Eigenvalue Problem (QEP) is a generalization of the standard eigenvalue problem that involves a quadratic eigenvalue operator. It seeks to find the eigenvalues and eigenvectors of the form: \[ A \lambda^2 + B \lambda + C = 0 \] where \(A\), \(B\), and \(C\) are given matrices, \(\lambda\) is the eigenvalue, and \(x\) is the corresponding eigenvector.

Squeeze mapping is likely a term related to methods used in various fields such as data visualization, machine learning, or statistics, but it may not be a standard term in widely recognized literature. Here are a few contexts where similar concepts may be applied: 1. **Data Visualization**: In data visualization, "squeeze" could refer to techniques used to compress or manipulate data representations to highlight certain patterns or trends. This could involve reducing the scale of a data set to make it easier to interpret.

In linear algebra, a **quotient space** is a way to construct a new vector space from an existing vector space by partitioning it into equivalence classes. This process can be thought of as "modding out" by a subspace, leading to a new space that captures certain properties while ignoring others.

Rank-width is a graph parameter that measures the complexity of a graph in terms of linear algebraic properties. It is defined in terms of the ranks of the adjacency matrix of the graph. More formally, the rank-width of a graph \( G \) can be understood through a specific type of tree decomposition.

In linear algebra, the **rank** of a matrix is defined as the maximum number of linearly independent row vectors or column vectors in the matrix. In simpler terms, it provides a measure of the "dimension" of the vector space spanned by its rows or columns.

Rank factorization is a mathematical concept that deals with the representation of a matrix as the product of two or more matrices. Specifically, it involves decomposing a matrix into factors that can provide insights into its structure and properties, particularly concerning the rank.

Reducing subspace, often referred to in the context of dimensionality reduction in fields such as machine learning and statistics, typically refers to a lower-dimensional representation of data that retains the essential characteristics of the original high-dimensional space. The main goal of reducing subspaces is to simplify the data while preserving relevant information, allowing for more efficient computation, enhanced visualization, or improved performance on specific tasks.

Regularized Least Squares is a variant of the standard least squares method used for linear regression that incorporates regularization techniques to prevent overfitting, especially in situations where the model might become too complex relative to the amount of available data. The standard least squares objective function minimizes the sum of the squared differences between observed values and predicted values.

The Sherman-Morrison formula is a statement in linear algebra that provides a way to compute the inverse of a matrix when that matrix is modified by the addition of a rank-one update.

A signal-flow graph (SFG) is a graphical representation used in control system engineering and signal processing to illustrate the flow of signals through a system. It represents the relationships between variables in a system, allowing for an intuitive understanding of how inputs are transformed into outputs through various paths. Here are the key components and features of a signal-flow graph: 1. **Nodes**: Represent system variables (such as system inputs, outputs, and intermediate signals). Each node corresponds to a variable in the system.

Ridge regression, also known as Tikhonov regularization, is a technique used in linear regression that introduces a regularization term to prevent overfitting and improve the model's generalization to new data. It is particularly useful when dealing with multicollinearity, where predictor variables are highly correlated.

Row equivalence is a concept in linear algebra that pertains to matrices. Two matrices are said to be row equivalent if one can be transformed into the other through a sequence of elementary row operations. These operations include: 1. **Row swapping**: Exchanging two rows of a matrix. 2. **Row scaling**: Multiplying all entries in a row by a non-zero scalar. 3. **Row addition**: Adding a multiple of one row to another row.

The Rule of Sarrus is a mnemonic used to evaluate the determinant of a \(3 \times 3\) matrix. It is particularly useful because it provides a simple and intuitive way to compute the determinant without resorting to the more formal cofactor expansion method.

The S-procedure is a mathematical technique used in convex optimization and control theory, specifically in the context of robust control and system stability analysis. It provides a way to transform certain types of inequalities involving quadratic forms into conditions that can be expressed in terms of linear matrix inequalities (LMIs).

Singular Value Decomposition (SVD) is a mathematical technique in linear algebra used to factorize a matrix into three other matrices. It is particularly useful for analyzing and reducing the dimensionality of data, solving linear equations, and performing principal component analysis.

The Special Linear Group, commonly denoted as \( \text{SL}(n, \mathbb{F}) \), is a fundamental concept in linear algebra and group theory. It consists of all \( n \times n \) matrices with entries from a field \( \mathbb{F} \) that have a determinant equal to 1.

The Spectral Theorem is a fundamental result in linear algebra and functional analysis that pertains to the diagonalization of certain types of matrices and operators. It provides a relationship between a linear operator or matrix and its eigenvalues and eigenvectors.

Spherical basis refers to a coordinate system or basis set defined for mathematical or physical problems, particularly in fields such as quantum mechanics, electromagnetism, and other areas of physics and engineering. The spherical basis is particularly useful for problems that are inherently spherically symmetric. ### Characteristics of Spherical Basis 1. **Coordinates**: The spherical basis is typically defined in terms of three coordinates: - \( r \): the radial distance from the origin.

Spinors are mathematical objects used in physics and mathematics, particularly in the context of quantum mechanics and the theory of relativity. In three dimensions, spinors can be understood as a generalization of the notion of vectors and can be associated with the representation of the rotation group, specifically the special orthogonal group SO(3). ### Definition and Representation In three-dimensional space, spinors are typically expressed in relation to the group of rotations SO(3).

Split-complex numbers, also known as hyperbolic numbers or null numbers, are a type of number that extends the real numbers similarly to how complex numbers extend them. They are defined as numbers of the form: \[ z = x + yj \] where \( x \) and \( y \) are real numbers, and \( j \) is a unit with the property that \( j^2 = 1 \).