Square root of a matrix

In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix product BB is equal to A. In general, a matrix can have several square roots. For example, the matrix ( 33 24 48 57 ) { extstyle left({egin{smallmatrix}33&24\48&57end{smallmatrix}} ight)} has square roots ( 1 4 8 5 ) { extstyle left({egin{smallmatrix}1&4\8&5end{smallmatrix}} ight)} and ( 5 2 4 7 ) { extstyle left({egin{smallmatrix}5&2\4&7end{smallmatrix}} ight)} , as well as their additive inverses. Another example is the 2×2 identity matrix ( 1 0 0 1 ) , {displaystyle left({egin{smallmatrix}1&0\0&1end{smallmatrix}} ight),} which has infinitely many symmetric rational square roots given by where ( r , s , t ) {displaystyle (r,s,t)} is any Pythagorean triple—that is, any set of positive integers such that r 2 + s 2 = t 2 {displaystyle r^{2}+s^{2}=t^{2}} . 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 non-integral. For example, the matrix ( 0 4 − 1 5 ) {displaystyle left({egin{smallmatrix}0&4\-1&5end{smallmatrix}} ight)} has the non-integer square root ( 2 3 4 3 − 1 3 7 3 ) {displaystyle left({egin{smallmatrix}{frac {2}{3}}&{frac {4}{3}}\-{frac {1}{3}}&{frac {7}{3}}end{smallmatrix}} ight)} as well as the integer square root matrix ( 2 − 4 1 − 3 ) {displaystyle left({egin{smallmatrix}2&-4\1&-3end{smallmatrix}} ight)} .The 2×2 identity matrix is another example of a matrix with a square root whose entries are rational, yet non-integral. A 2×2 matrix with two distinct nonzero eigenvalues has four square roots. More generally, an n×n matrix with n distinct nonzero eigenvalues has 2n square roots. Such a matrix, A, has a decomposition VDV−1 where V is the matrix whose columns are eigenvectors of A and D is the diagonal matrix whose diagonal elements are the corresponding n eigenvalues λi. Thus the square roots of A are given by VD½ V−1, where D½ is any square root matrix of D, which, for distinct eigenvalues, must be diagonal with diagonal elements equal to square roots of the diagonal elements of D; since there are two possible choices for a square root of each diagonal element of D, there are 2n choices for the matrix D½.