characteristic root of a square matrix

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Explanation of "Characteristic Root of a Square Matrix"

Definition: In mathematics, particularly in linear algebra, the "characteristic root" of a square matrix is a special number. It is found by taking a square matrix (a grid of numbers with the same number of rows and columns) and subtracting a certain number (the characteristic root) multiplied by the identity matrix (a special kind of matrix that behaves like the number 1 in multiplication). When you do this and the result is a matrix that has a determinant (a calculated value that can tell us important things about the matrix) equal to zero, the number you subtracted is called a characteristic root.

Usage Instructions:

• You typically use this term when studying properties of matrices, especially when solving systems of equations or understanding transformations in geometry.
• It's a key concept in higher mathematics, especially in calculus and algebra.

Example:

Imagine you have a square matrix A:

Advanced Usage:

• In more advanced mathematics, characteristic roots are often referred to as "eigenvalues." They are used in various fields, including physics and engineering, to analyze systems and solve differential equations.
• The process to find these roots is known as finding the "eigenvalues" and can be crucial for understanding the behavior of dynamic systems.

Word Variants:

• Characteristic Value: Another term often used interchangeably with characteristic root.
• Eigenvalue: A more advanced term that is commonly used in mathematics and related fields.

Different Meanings:

• In other contexts, "characteristic" can refer to features or traits of a person or object (e.g., "He has a characteristic smile" means that smile is a typical feature of that person).

Synonyms:

• Eigenvalue (in advanced contexts)
• Characteristic Value

Idioms and Phrasal Verbs:

While the term "characteristic root of a square matrix" is very specific to mathematics, here are a few related idioms and phrases in a broader context: - "Root of the problem": This idiom means to identify the fundamental cause of an issue, not related to mathematics but can be used in problem-solving discussions. - "Get to the root": This phrase means to find the core issue or main point of something.

Summary:

The "characteristic root of a square matrix" is a mathematical concept that identifies special numbers related to the properties of matrices.

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
• eigenvalue of a square matrix