Children: Linear algebra
Translation of axes refers to the process of moving a coordinate system along its axes without rotation. This involves shifting the origin of the coordinate system to a new location in the same dimensional space, effectively changing the coordinates of points relative to the new origin. In a two-dimensional Cartesian coordinate system, for instance, translating the axes means moving both the x-axis and the y-axis by a certain distance in the same direction.
In the context of linear algebra, the transpose of a linear map is a fundamental concept that relates to how linear transformations interact with dual spaces. ### Definition Let \( T: V \to W \) be a linear map between two finite-dimensional vector spaces \( V \) and \( W \).
The triangle inequality is a fundamental concept in geometry and mathematics that states the following for any triangle with sides of lengths \( a \), \( b \), and \( c \): 1. \( a + b > c \) 2. \( a + c > b \) 3. \( b + c > a \) In essence, the triangle inequality asserts that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
A unitary transformation is a mathematical operation that transforms a quantum state in a Hilbert space while preserving the inner product, and, consequently, the probabilities associated with quantum measurements. In more formal terms, if you have a quantum state \( | \psi \rangle \), a unitary transformation \( U \) acts on this state to produce a new state \( | \psi' \rangle = U | \psi \rangle \).
Vectorization in mathematics, particularly in the context of linear algebra and computational mathematics, refers to the process of converting an operation that is typically performed on scalars or a collection of operations on individual elements into an operation that can be applied to vectors or matrices in a more efficient and compact form. This technique is often used to enhance performance in numerical computations, particularly in programming environments that support vectorized operations, such as NumPy in Python or MATLAB.
Weyl's inequality is a result in linear algebra and matrix theory concerning the eigenvalues of Hermitian (or symmetric) matrices. It relates the eigenvalues of the sum of two Hermitian matrices to the eigenvalues of the individual matrices. Let's denote two Hermitian matrices \( A \) and \( B \).
Weyr canonical form is a representation of a matrix that displays its structure in a standardized way, similar to Jordan canonical form, but with some differences. It specifically relates to the eigenvalues and the generalized eigenvectors of a matrix, particularly in the context of linear algebra. In the Weyr canonical form, the matrix is represented in a way that organizes the eigenvalues and their corresponding generalized eigenvectors into blocks.
A Z-order curve, also known as a Z-ordering or Morton order, is a spatial filling curve that is used to map multi-dimensional data (like two-dimensional coordinates) into one-dimensional data while preserving the spatial locality of the points. This means that points that are close together in the multi-dimensional space will remain close together in the one-dimensional representation. The Z-ordering works by interleaving the binary representations of the coordinates of the points.
The Zassenhaus algorithm is an algorithm used for factoring integers, particularly effective for finding the prime factors of integers that are the product of two large primes. It was developed by Hans Zassenhaus in the 1980s and is notable for its application in computational number theory and cryptography. The algorithm incorporates several techniques and concepts, including: 1. **Quadratic Sieve**: It employs a number-theoretic sieve method to identify and collect potential factors.
Zech's logarithm, denoted as \( z \), is a mathematical construct used primarily in the field of finite fields and combinatorial structures, such as in coding theory and cryptography. It arises in relation to the concepts of logarithms in finite fields, specifically in the context of operations involving powers of elements in these fields.