with a matrix of full row rank (i.e. of rank ). Moreover, for any decomposition , . The columns of can be seen as vectors in the complex or real vector space , respectively. Then the entries of are inner products (that is dot products, in the real case) of these vectors In other words, a Hermitian matrix is positive semidefinite if and only if it is the Gram matrix of some vectors . It is positive definite if and only if it is the Gram matrix of some linearly independent vectors. In general, the rank of the Gram matrix of vectors equals the dimension of the space spanned by these vectors. Uniqueness up to unitary transformations The decomposition is not unique: if for some matrix and if is any unitary matrix (meaning ), then for . However, this is the only way in which two decompositions can differ: the decomposition is unique up to unitary transformations. More formally, if is a matrix and is a matrix such that , then there is a matrix with orthonormal columns (meaning ) such that . When this means is unitary. This statement has an intuitive geometric interpretation in the real case: let the columns of and be the vectors and in . A real unitary matrix is an orthogonal matrix, which describes a rigid transformation (an isometry of Euclidean space ) preserving the 0 point (i.e. rotations and reflections, without translations). Therefore, the dot products and are equal if and only if some rigid transformation of transforms the vectors to (and 0 to 0). Square root A Hermitian matrix is positive semidefinite if and only if there is a positive semidefinite matrix (in particular is Hermitian, so ) satisfying . This matrix is unique, is called the non-negative square root of , and is denoted with . When is positive definite, so is , hence it is also called the positive square root of . The non-negative square root should not be confused with other decompositions . Some authors use the name square root and for any such decomposition, or specifically for the Cholesky decomposition, or any decomposition of the form ; other only use it for the non-negative square root. If then . Cholesky decomposition A positive semidefinite matrix can be written as , where is lower triangular with non-negative diagonal (equivalently where is upper triangular); this is the Cholesky decomposition.