Editing Definite matrix (section)

=== Uniqueness up to unitary transformations ===
The decomposition is not unique: 
if <math>M = B^* B</math> for some <math>k \times n</math> matrix <math>B</math> and if <math>Q</math> is any [[unitary matrix|unitary]] <math>k \times k</math> matrix (meaning <math>Q^* Q = Q Q^* = I</math>),
then <math>M = B^* B = B^* Q^* Q B = A^* A</math> for <math>A = Q B.</math>

However, this is the only way in which two decompositions can differ: The decomposition is unique up to [[unitary transformation]]s.
More formally, if <math>A</math> is a <math>k \times n</math> matrix and <math>B</math> is a <math>\ell \times n</math> matrix such that <math>A^* A = B^* B,</math>
then there is a <math>\ell \times k</math> matrix <math>Q</math> with orthonormal columns (meaning <math>Q^* Q = I_{k \times k}</math>) such that <math>B = Q A.</math><ref>{{harvtxt|Horn|Johnson|2013}}, p. 452, Theorem 7.3.11</ref>
When <math>\ell = k</math> this means <math>Q</math> is [[unitary matrix|unitary]].

This statement has an intuitive geometric interpretation in the real case:
let the columns of <math>A</math> and <math>B</math> be the vectors <math>a_1,\dots,a_n</math> and <math>b_1, \dots, b_n</math> in <math>\mathbb{R}^k.</math>
A real unitary matrix is an [[orthogonal matrix]], which describes a [[rigid transformation]] (an isometry of Euclidean space <math>\mathbb{R}^k</math>) preserving the 0 point (i.e. [[Rotation matrix|rotations]] and [[Reflection matrix|reflections]], without translations). 
Therefore, the dot products <math>a_i \cdot a_j</math> and <math>b_i \cdot b_j</math> are equal if and only if some rigid transformation of <math>\mathbb{R}^k</math> transforms the vectors <math>a_1,\dots,a_n</math> to <math>b_1,\dots,b_n</math> (and 0 to 0).