eigenvalue of a square matrix

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Explanation of "Eigenvalue of a Square Matrix"

Definition: In simple terms, an eigenvalue is a special number related to a square matrix in mathematics. A square matrix is a grid of numbers with the same number of rows and columns. An eigenvalue is a number that helps us understand how that matrix transforms certain special vectors (called eigenvectors).

Usage Instructions:

• In Mathematics: You will often come across eigenvalues in subjects like linear algebra, physics, and engineering.
• Finding Eigenvalues: To find eigenvalues, you usually solve a specific equation called the characteristic equation of the matrix.

Example:

Let’s say we have a square matrix:

Advanced Usage:

• Application: Eigenvalues are used in various fields, including computer science for algorithms (like Google's PageRank), physics for stability analysis, and statistics in principal component analysis (PCA).
• Eigenvectors: Each eigenvalue has a corresponding eigenvector, which describes the direction of the transformation represented by the matrix.

Word Variants:

• Eigenvector: The vector associated with an eigenvalue that gives the direction in which the transformation acts.
• Eigenvalue Problem: The task of finding the eigenvalues and eigenvectors of a matrix.

Different Meaning:

In mathematics, "eigen" comes from German, meaning "own" or "self". So, it relates to the idea that these values (eigenvalues) are intrinsic to the matrix itself.

Synonyms:

There are no direct synonyms for "eigenvalue" in mathematics, as it has a specific meaning, but it may sometimes be discussed in the context of "characteristic value" in literature.

Idioms and Phrasal Verbs:

There are no idioms or phrasal verbs specifically associated with "eigenvalue," as it is a technical term used primarily within mathematics.

Conclusion:

Understanding eigenvalues is crucial for deeper studies in mathematics and its applications.

Academic

Noun

1. (mathematics) any number such that a given square matrix minus that number times the identity matrix has a zero determinant

Synonyms

• eigenvalue
• eigenvalue of a matrix
• characteristic root of a square matrix