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Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.
On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of A associated with λ. So, the set E is the union of the zero vector with the set of all eigenvectors of A associated with λ, and E equals the nullspace of (A − λI). E is called the eigenspace or characteristic space of A associated with λ.
Yet another usage is latent semantic indexing in natural-language text processing. In general numerical computation involving linear or linearized systems, there is a universal constant that characterizes the regularity or singularity of a problem, which is the system's "condition number" κ := σ max / σ min {\displaystyle \kappa :=\sigma ...
For a normal matrix A (and only for a normal matrix), the eigenvectors can also be made orthonormal (=) and the eigendecomposition reads as =. In particular all unitary , Hermitian , or skew-Hermitian (in the real-valued case, all orthogonal , symmetric , or skew-symmetric , respectively) matrices are normal and therefore possess this property.
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where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning =) and R is an upper triangular matrix (also called right triangular matrix). If A is invertible , then the factorization is unique if we require the diagonal elements of R to be positive.
The eigendecomposition (or spectral decomposition) of a diagonalizable matrix is a decomposition of a diagonalizable matrix into a specific canonical form whereby the matrix is represented in terms of its eigenvalues and eigenvectors. The spectral radius of a square matrix is the largest absolute value of its eigenvalues.
Spectral decomposition for matrix: eigendecomposition of a matrix Spectral decomposition for linear operator: spectral theorem Decomposition of spectrum (functional analysis)