Properties
In general, a matrix can have many 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 an infinitude of symmetric rational square roots given by and where (r, s, t) is any Pythagorean triple — that is, any set of positive integers such that
However, a positive-definite matrix has precisely one positive-definite 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 integral. For example, the matrix has the non-integer square root, as well as the integer 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 n×n matrix with n distinct nonzero eigenvalues has square roots. This is because such a matrix A has a decomposition where P is the matrix whose columns are eigenvectors of A and D is the diagonal matrix whose diagonal elements are the corresponding eigenvalues . Thus the square roots of A are given by where is any square root matrix of D, which 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 choices of the matrix . 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, 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.
Read more about this topic: Square Root Of A Matrix
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)