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In numerical linear algebra, the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix.The QR algorithm was developed in the late 1950s by John G. F. Francis and by Vera N. Kublanovskaya, working independently.
A basic minor of a matrix is the determinant of a square submatrix that is of maximal size with nonzero determinant. [3] For Hermitian matrices, the leading principal minors can be used to test for positive definiteness and the principal minors can be used to test for positive semidefiniteness. See Sylvester's criterion for more details.
Principal component analysis of the correlation matrix provides an orthogonal basis for the space of the observed data: In this basis, the largest eigenvalues correspond to the principal components that are associated with most of the covariability among a number of observed data.
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 ...
In particular, the diagonal entries are the principal minors of , which of course are also principal minors of , and are thus non-negative. Since the trace of a matrix is the sum of the diagonal entries, it follows that tr ( ⋀ j M k ) ≥ 0. {\displaystyle \operatorname {tr} \left(\textstyle \bigwedge ^{j}M_{k}\right)\geq 0.}
The eigenvalues of each principal submatrix of the Hessenberg operator are given by the characteristic polynomial for that submatrix. These polynomials are called the Bergman polynomials, and provide an orthogonal polynomial basis for Bergman space.
The eigenvalues and eigenvectors of are simply those of and , and it will almost always be faster to solve these two smaller problems than to solve the original problem all at once. This technique can be used to improve the efficiency of many eigenvalue algorithms, but it has special significance to divide-and-conquer.
In mathematics, a totally positive matrix is a square matrix in which all the minors are positive: that is, the determinant of every square submatrix is a positive number. [1] A totally positive matrix has all entries positive, so it is also a positive matrix; and it has all principal minors positive (and positive eigenvalues).