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Concept
Square root of a matrix
Summary
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.
Some authors use the name square root or the notation A1/2 only for the specific case when A is positive semidefinite, to denote the unique matrix B that is positive semidefinite and such that BB = BTB = A (for real-valued matrices, where BT is the transpose of B).
Less frequently, the name square root may be used for any factorization of a positive semidefinite matrix A as BTB = A, as in the Cholesky factorization, even if BB ≠ A. This distinct meaning is discussed in .
In general, a matrix can have several square roots. In particular, if then as well.
The 2×2 identity matrix has infinitely many square roots. They are given by
and
where are any numbers (real or complex) such that .
In particular if is any Pythagorean triple—that is, any set of positive integers such that , then
is a square root matrix of which is symmetric and has rational entries.
Thus
Minus identity has a square root, for example:
which can be used to represent the imaginary unit i and hence all complex numbers using 2×2 real matrices, see Matrix representation of complex numbers.
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
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, as in examples above.
Positive definite matrix#Decomposition
A symmetric real n × n matrix is called positive semidefinite if for all (here denotes the transpose, changing a column vector x into a row vector).
A square real matrix is positive semidefinite if and only if for some matrix B.
There can be many different such matrices B.
A positive semidefinite matrix A can also have many matrices B such that .
Official source
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