The existence of a square root of a singular matrix depends on the Jordan structure of the zero eigenvalues. If is nonsingular then it has at least square roots, where is the number of distinct eigenvalues. Indeed if is symmetric positive definite then it has a spectral decomposition, where is orthogonal and is diagonal with positive diagonal elements, and then is also symmetric positive definite. Ī symmetric positive definite matrix has a unique symmetric positive definite square root. If is nonsingular and has no eigenvalues on the negative real axis then has a unique principal square root. The matrix square root of most practical interest is the one whose eigenvalues lie in the right half-plane, which is called the principal square root, written. Clearly, a square root of a diagonal matrix need not be diagonal. Has infinitely many square roots (namely the involutory matrices), including, the lower triangular matrixĪnd any symmetric orthogonal matrix, such as For, depending on the matrix there can be no square roots, finitely many, or infinitely many. įor a scalar ( ), there are two square roots (which are equal if ), and they are real if and only if is real and nonnegative. A square root of an matrix is any matrix such that.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |