diagonalize
Verb: 1. To transform a matrix into a diagonal matrix: In linear algebra, to perform an operation on a square matrix so that it becomes a diagonal matrix, meaning all entries outside the main diagonal are zero. This is typically achieved by finding a basis of eigenvectors.
The verb "diagonalize" is used in the context of mathematics, specifically linear algebra. It describes a specific computational or theoretical process applied to a matrix. * A matrix that can be transformed in this way is described as diagonalizable. * The process itself is called diagonalization.
- "The goal of the exercise is to diagonalize the given 3x3 matrix."
- "Not all matrices can be diagonalized; only those with a sufficient number of linearly independent eigenvectors."
- "After we diagonalize the system's matrix, solving the differential equations becomes much simpler."
- Diagonalization Theorem: A fundamental theorem stating conditions under which a matrix is diagonalizable.
- Example: "The Diagonalization Theorem provides the criteria for when a matrix A can be diagonalized."
- Simultaneous Diagonalization: The process of diagonalizing two or more matrices with the same eigenvector basis.
- Example: "Commuting matrices can often undergo simultaneous diagonalization."
- Diagonalizable (adjective): Describing a matrix that can be diagonalized.
- Example: "A symmetric matrix is always diagonalizable."
- Diagonalization (noun): The process or result of diagonalizing.
- Example: "The diagonalization of the matrix revealed its eigenvalues on the main diagonal."
- Diagonal Matrix (noun): The resulting matrix after diagonalization, where all off-diagonal entries are zero.
- Transform to diagonal form: A more descriptive phrase with the same meaning.
As a technical mathematical term, "diagonalize" does not have phrasal verbs. Its usage is confined to its specific definition. * Eigenvalue: A scalar associated with a linear transformation that appears on the diagonal of a diagonalized matrix. * Eigenvector: A non-zero vector that changes only by a scalar factor when a linear transformation is applied; these form the basis for diagonalization. * Similarity Transformation: The operation P⁻¹AP used to diagonalize a matrix A, where P is the matrix of eigenvectors.
- transform a matrix to a diagonal matrix