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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.
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.
Let = be an positive matrix: > for ,.Then the following statements hold. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue (also called the leading eigenvalue, principal eigenvalue or dominant eigenvalue), such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) in absolute value is strictly smaller than r, |λ| < r.
Hurwitz matrix: A matrix whose eigenvalues have strictly negative real part. A stable system of differential equations may be represented by a Hurwitz matrix. M-matrix: A Z-matrix with eigenvalues whose real parts are nonnegative. Positive-definite matrix: A Hermitian matrix with every eigenvalue positive. Stability matrix: Synonym for Hurwitz ...
By means of this polynomial, determinants can be used to find the eigenvalues of the matrix : they are precisely the roots of this polynomial, i.e., those complex numbers such that = A Hermitian matrix is positive definite if all its eigenvalues are positive.
In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector, where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...
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 ...
If is positive definite, then the eigenvalues are all positive reals, so the chosen diagonal of also consists of positive reals. Hence the eigenvalues of Q U 1 2 Q ∗ {\displaystyle QU^{\frac {1}{2}}Q^{*}} are positive reals, which means the resulting matrix is the principal root of A {\displaystyle A} .