Live briefing
Everything about Diagonalizable matrix
Matrices similar to diagonal matrices
In linear algebra, a square matrix is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists an invertible matrix and a diagonal matrix such that. This is equivalent to . This property exists for any linear map: for a finite-dimensional vector space , a linear map is called diagonalizable if there exists an ordered basis of consisting of eigenvectors of. These definitions are equivalent: if has a matrix representation as above, then the column vectors of form a basis consisting of eigenvectors of , and the diagonal entries of are the corresponding eigenvalues of ; with respect to this eigenvector basis, is represented by . Wikipedia â
đ§ Start here: a 6-step reading path
- Read the overview: Diagonalizable matrixA 5-minute orientation
- Skim a recent paper: Solutions of the YangâBaxterâlike matrix equation with 3Ă3 diagonalizable coefficient matrixInfluential work since 2022
- Pick up a classic book: Coding the MatrixThe most-reprinted book on the subject
- Read the landmark paper: Complete commuting solutions of the YangâBaxter-like matrix equation for diagonalizable matricesCited 40 times
- Go sideways: Spectral theoremA closely related topic
- Go sideways: Jordan normal formA closely related topic
đ Definitions
No dictionary entry for this exact phrase.
đ Landmark papers (most cited)
Complete commuting solutions of the YangâBaxter-like matrix equation for diagonalizable matrices
Explicit solutions of the Yang-Baxter-like matrix equation for a singular diagonalizable matrix with three distinct eigenvalues
The Probability that a Matrix of Integers Is Diagonalizable
All solutions of the YangâBaxter-like matrix equation for diagonalizable coefficient matrix with two different eigenvalues
On maps preserving products equal to a diagonalizable matrix
A family of explicitly diagonalizable weighted Hankel matrices generalizing the Hilbert matrix
Finding Solutions to the YangâBaxter-like Matrix Equation for Diagonalizable Coefficient Matrix
Explicit solutions of the YangâBaxter-like matrix equation for a diagonalizable matrix with spectrum contained in {1, α, 0}
đ Influential since 2022
Solutions of the YangâBaxterâlike matrix equation with 3Ă3 diagonalizable coefficient matrix
Applications of unitarily diagonalizable matrices in an indefinite inner product space to matrix partial orders
Remarks on the controllability of parabolic systems with non-diagonalizable diffusion matrix
Explicit solutions of the Yang-Baxter-like matrix equation for diagonalizable coefficient matrix with two distinct nonzero eigenvalues
Solving the Yang-Baxter-like matrix equation for diagonalizable coefficient matrix when A^3=-A
đ Books worth reading
Coding the MatrixPhilip N. Klein · 2013
Jordan canonical formSteven H. Weintraub · 2008
Principles of linear algebra with MathematicaKenneth Shiskowski · 2011- đProceedings of the International Congress of Mathematicians Madrid, August 22-30, 2006Marta Sanz-SolĂ©, Javier Soria · 2007
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
đ Related topics
- Spectral theorem In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, representedâŠ
- Jordan normal form [âλ11âλ11âλ1ââλ21ââλ2â[λ3]â±âλn1ââλnâ] Example of a matrix in Jordan normal form. All matrix entries not shown are zero. The outlined squares are knownâŠ
- Eigenvalues and eigenvectors In linear algebra, an eigenvector (/ËaÉȘÉĄÉn-/ EYE-gÉn-) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) byâŠ
- Eigendecomposition of a matrix In linear algebra, eigendecomposition (also known as eigenvalue decomposition or EVD) is a factorization of a matrix A into a canonical form given by â A=QDQâđŁâ âŠ
- Normal matrix In mathematics, a complex square matrix A is normal if it commutes with its conjugate transpose A*: A normalâșAâA=AAâ. The concept of normal matrices canâŠ
- Diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matricesâŠ
- Compact operator on Hilbert space In the mathematical discipline of functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix actingâŠ
- Generalized eigenvector In linear algebra, a generalized eigenvector of an nĂn matrix A is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary)âŠ