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In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. Matrix is said to be a square root of if the matrix product is equal to . ==Properties== In general, a matrix can have several square roots. For example, the matrix has square roots and , as well as their additive inverses. Another example is the 2×2 identity matrix which has infinitely many symmetric rational square roots given by : where is any Pythagorean triple—that is, any set of positive integers such that .〔Mitchell, Douglas W. "Using Pythagorean triples to generate square roots of ''I''2". ''The Mathematical Gazette'' 87, November 2003, 499-500.〕 However, a ''positive-semidefinite matrix'' has precisely one positive-semidefinite square root, which can be called its ''principal square root''. While the square root of a nonnegative integer is either again an integer or an irrational number, in contrast an ''integer matrix'' can have a square root whose entries are rational, yet not integer. For example, the matrix has the non-integer square root as well as the integer square root matrix . The 2×2 identity matrix is another example. A 2×2 matrix with two distinct nonzero eigenvalues has four square roots. More generally, an matrix with ''distinct nonzero eigenvalues'' has 2n square roots. Such a matrix, , has a decomposition where is the matrix whose columns are eigenvectors of and is the diagonal matrix whose diagonal elements are the corresponding eigenvalues . Thus the square roots of are given by , where ½ is any square root matrix of , which, for distinct eigenvalues, must be diagonal with diagonal elements equal to square roots of the diagonal elements of ; since there are two possible choices for a square root of each diagonal element of , there are 2''n'' choices for the matrix ½. This also leads to a proof of the above observation, that a positive-definite matrix has precisely one positive-definite square root: a positive definite matrix has only positive eigenvalues, and each of these eigenvalues has only one positive square root; and since the eigenvalues of the square root matrix are the diagonal elements of ½, for the square root matrix to be itself positive definite necessitates the use of only the unique positive square roots of the original eigenvalues. Just as with the real numbers, a real matrix may fail to have a real square root, but have a square root with complex-valued entries. Some matrices have ''no square root''. An example is the matrix . In general, a complex matrix with positive real eigenvalues has a unique square root with positive eigenvalues called the ''principal square root''. Moreover, the operation of taking the principal square root is continuous on this set of matrices. If the matrix has real entries, then the square root also has real entries. These properties are consequences of the holomorphic functional calculus applied to matrices. The existence and uniqueness of the principal square root can be deduced directly from the Jordan normal form (see below).〔For analytic functions of matrices, see * *〕 〔For the holomorphic functional calculus, see: * * * 〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「square root of a matrix」の詳細全文を読む スポンサード リンク
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