Part of Speech

Noun

Phonetic Transcription

/ədˈdʒɔɪnt ˈmeɪtrɪks/

Meaning and Usage

An adjoint matrix is a matrix derived from another matrix by taking the transpose of the cofactor matrix. It plays a crucial role in linear algebra, particularly in calculating the inverse of a matrix and understanding properties of matrices in relation to linear transformations.

The term is primarily used in written contexts, particularly in textbooks, research papers, and academic settings, rather than in everyday oral speech. It is often encountered in mathematics, physics, engineering, and computer science.

Example Sentences

1. The adjoint matrix can be used to find the inverse of a matrix when the determinant is non-zero.

2. Translation: La matriz adjunta se puede usar para encontrar la inversa de una matriz cuando el determinante no es cero.

3. In linear algebra, calculating the adjoint matrix is an essential step in solving systems of equations.

4. Translation: En álgebra lineal, calcular la matriz adjunta es un paso esencial para resolver sistemas de ecuaciones.

5. The properties of the adjoint matrix make it useful in various applications, such as optimization and control theory.

6. Translation: Las propiedades de la matriz adjunta la hacen útil en diversas aplicaciones, como la optimización y la teoría de control.

Idiomatic Expressions

The term "adjoint" may not have extensive idiomatic expressions uniquely associated with it. However, it frequently appears in mathematical phrases and theories related to matrices. Below are some expressions in context:

1. The adjoint of a matrix is critical for determining eigenvectors.

2. Translation: La adjunta de una matriz es crítica para determinar los eigenvectores.

3. To compute the adjoint of a matrix, one must first find the cofactors of each element.

4. Translation: Para calcular la adjunta de una matriz, primero hay que encontrar los cofactores de cada elemento.

5. Often, the adjoint of a matrix is used to simplify complex equations in linear systems.

6. Translation: A menudo, la adjunta de una matriz se utiliza para simplificar ecuaciones complejas en sistemas lineales.

Etymology

The term "adjoint" originates from the Latin word "adiungere," which means "to join to," reflecting its mathematical function of being related to another matrix. The term has been adopted in many branches of mathematics and retains its roots in ideas of joining and connection.

Synonyms and Antonyms

Synonyms: - Classical adjoint - Adjoint operator (in certain contexts)

Antonyms: There are no direct antonyms for "adjoint matrix" since it is a specific mathematical term. However, one could argue that the concept of a "non-adjoint matrix" could serve as a contextual opposite when comparing properties and applications.