Defective matrix (original) (raw)
En àlgebra lineal, una matriu defectiva és una matriu quadrada que no té una base completa de vectors propis, i és per això no diagonalizable. En particular, una matriu n × n és defectiva si i només si no té n vectors propis linealment independents. Es crea una base completa augmentant els vectors propis amb vectors propis generalitzats, que són necessaris per solucionar sistemes defectius d'equacions diferencials ordinàries i altres problemes.
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| dbo:abstract | En àlgebra lineal, una matriu defectiva és una matriu quadrada que no té una base completa de vectors propis, i és per això no diagonalizable. En particular, una matriu n × n és defectiva si i només si no té n vectors propis linealment independents. Es crea una base completa augmentant els vectors propis amb vectors propis generalitzats, que són necessaris per solucionar sistemes defectius d'equacions diferencials ordinàries i altres problemes. Una matriu n × n defectiva sempre té menys que n valors propis diferents, ja que quan els valors propis són diferents tenen vectors propis linealment independents. En particular, una matriu defectiva té un o més valors propis λ amb multiplicitat algebraica m > 1 (és a dir, les arrels del seu polinomi característic són múltiples), però menys d'm vectors propis linealment independents associats a λ. Si la multiplicitat algebraica de λ supera la seva multiplicitat geomètrica (és a dir, el número de vectors propis linealment independents associats a λ), llavors λ és anomenat valor propi defectiu. Tanmateix, cada valor propi amb multiplicitat algebraica m sempre té m vectors propis generalitzats independents. Una matriu hermítica (o el cas particular en els reals d'una matriu simètrica) o una matriu unitària mai és defectiva. Més generalment, una matriu normal (quin inclou l'hermítica i la unitària com a casos especials) mai és defectiva. (ca) In linear algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable. In particular, an n × n matrix is defective if and only if it does not have n linearly independent eigenvectors. A complete basis is formed by augmenting the eigenvectors with generalized eigenvectors, which are necessary for solving defective systems of ordinary differential equations and other problems. An n × n defective matrix always has fewer than n distinct eigenvalues, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues λ with algebraic multiplicity m > 1 (that is, they are multiple roots of the characteristic polynomial), but fewer than m linearly independent eigenvectors associated with λ. If the algebraic multiplicity of λ exceeds its geometric multiplicity (that is, the number of linearly independent eigenvectors associated with λ), then λ is said to be a defective eigenvalue. However, every eigenvalue with algebraic multiplicity m always has m linearly independent generalized eigenvectors. A Hermitian matrix (or the special case of a real symmetric matrix) or a unitary matrix is never defective; more generally, a normal matrix (which includes Hermitian and unitary as special cases) is never defective. (en) |
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| rdfs:comment | En àlgebra lineal, una matriu defectiva és una matriu quadrada que no té una base completa de vectors propis, i és per això no diagonalizable. En particular, una matriu n × n és defectiva si i només si no té n vectors propis linealment independents. Es crea una base completa augmentant els vectors propis amb vectors propis generalitzats, que són necessaris per solucionar sistemes defectius d'equacions diferencials ordinàries i altres problemes. (ca) In linear algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable. In particular, an n × n matrix is defective if and only if it does not have n linearly independent eigenvectors. A complete basis is formed by augmenting the eigenvectors with generalized eigenvectors, which are necessary for solving defective systems of ordinary differential equations and other problems. (en) |
| rdfs:label | Matriu defectiva (ca) Defective matrix (en) |
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