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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.
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, ... is equal to the number of non-zero eigenvalues of ...
The eigenvalues of a Hermitian matrix are real, since (λ − λ)v = (A * − A)v = (A − A)v = 0 for a non-zero eigenvector v. If A is real, there is an orthonormal basis for R n consisting of eigenvectors of A if and only if A is symmetric. It is possible for a real or complex matrix to have all real eigenvalues without being Hermitian.
Also (in the maximum theorem) subsequent eigenvalues and eigenvectors are found by induction and orthogonal to each other; therefore, = with , =, <. The Courant minimax principle, as well as the maximum principle, can be visualized by imagining that if || x || = 1 is a hypersphere then the matrix A deforms that hypersphere into an ellipsoid .
In case of a symmetric matrix it is the absolute value of the quotient of the largest and smallest eigenvalue. Matrices with large condition numbers can cause numerically unstable results: small perturbation can result in large errors. Hilbert matrices are the most famous ill-conditioned matrices.
If the linear transformation is expressed in the form of an n by n matrix A, then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication =, where the eigenvector v is an n by 1 matrix. For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix—for example by diagonalizing it.
We will now discuss how these difficulties manifest in the basic QR algorithm. This is illustrated in Figure 2. Recall that the ellipses represent positive-definite symmetric matrices. As the two eigenvalues of the input matrix approach each other, the input ellipse changes into a circle. A circle corresponds to a multiple of the identity matrix.
There are two types of continuity concerning eigenvalues: (1) each individual eigenvalue is a usual continuous function (such a representation does exist on a real interval but may not exist on a complex domain), (2) eigenvalues are continuous as a whole in the topological sense (a mapping from the matrix space with metric induced by a norm to ...