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A 2×2 real and symmetric matrix representing a stretching and shearing of the plane. The eigenvectors of the matrix (red lines) are the two special directions such that every point on them will just slide on them. The example here, based on the Mona Lisa, provides a simple illustration. Each point on the painting can be represented as a vector ...
Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation [1] =,where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.l When k = 1, the vector is called simply an eigenvector, and the pair ...
Notation: The index j represents the jth eigenvalue or eigenvector. The index i represents the ith component of an eigenvector. Both i and j go from 1 to n, where the matrix is size n x n. Eigenvectors are normalized. The eigenvalues are ordered in descending order.
For example, the fourth-order Hilbert matrix has a condition of 15514, while for order 8 it is 2.7 × 10 8. Rank A matrix A {\displaystyle A} has rank r {\displaystyle r} if it has r {\displaystyle r} columns that are linearly independent while the remaining columns are linearly dependent on these.
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 {\displaystyle n\times n} matrix is defective if and only if it does not have n {\displaystyle n} linearly independent eigenvectors. [ 1 ]
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.
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The vector = (,,, …,) is an eigenvector of this matrix, where the eigenvalue is a root of (). Setting the initial values of the sequence equal to this vector produces a geometric sequence a k = λ k {\displaystyle a_{k}=\lambda ^{k}} which satisfies the recurrence.