DIAGONALIZABLE Meaning and
Definition
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Diagonalizable is an adjective used in mathematics to describe a square matrix that can be transformed into a diagonal matrix by using a similarity transformation. In other words, a matrix A is said to be diagonalizable if there exists an invertible matrix P and a diagonal matrix D such that A = PDP^(-1).
This definition implies that the columns of matrix P can diagonalize matrix A, as multiplying P by A and then multiplying the result by the inverse of P yields a diagonal matrix D. This process is often employed to simplify computations involving matrices, as diagonal matrices tend to be significantly more straightforward to manipulate and analyze.
For a matrix A to be considered diagonalizable, it must satisfy certain conditions. One condition is that A must have n linearly independent eigenvectors, where n is the order of the matrix. Additionally, if a matrix has a repeated eigenvalue but does not possess n linearly independent eigenvectors corresponding to that eigenvalue, then it cannot be diagonalized.
The concept of diagonalizability holds great significance in various areas of mathematics, particularly in linear algebra, differential equations, and optimization. It allows for the extraction of valuable information about the behavior and properties of matrices, enabling more efficient calculations and insights.
Common Misspellings for DIAGONALIZABLE
- siagonalizable
- xiagonalizable
- ciagonalizable
- fiagonalizable
- riagonalizable
- eiagonalizable
- duagonalizable
- djagonalizable
- dkagonalizable
- doagonalizable
- d9agonalizable
- d8agonalizable
- dizgonalizable
- disgonalizable
- diwgonalizable
- diqgonalizable
- diafonalizable
- diavonalizable
- diabonalizable
- diagonalizabel
- diaganalizible
Etymology of DIAGONALIZABLE
The term "diagonalizable" is derived from the word "diagonalize". word "diagonalize" is a combination of two words: "diagonal" and "-ize".
"Diagonal" comes from the Greek word "diágonios", which means "from angle to angle" or "across a corner". It is derived from the Greek words "diá", meaning "through" or "across", and "gonía", meaning "angle" or "corner".
The suffix "-ize" is a common English derivational suffix that is borrowed from Greek. It is used to form verbs, indicating the act, process, or result of making or becoming something.
Therefore, "diagonalize" refers to the action of making something or becoming something diagonal. In mathematics, "diagonalize" is specifically used to describe the process of transforming a matrix into a diagonal matrix through a change of basis.
Similar spelling words for DIAGONALIZABLE
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