Editing Cholesky decomposition (section)

=== Proof by QR decomposition ===

Let <math display=inline>\mathbf{A}</math> be a [[Positive-definite matrix|positive semi-definite]] Hermitian matrix. Then it can be written as a product of its [[Square root of a matrix|square root matrix]], <math display=inline>\mathbf{A} = \mathbf{B} \mathbf{B}^*</math>. Now [[QR decomposition]] can be applied to <math display=inline>\mathbf{B}^*</math>, resulting in <math display=inline>\mathbf{B}^* = \mathbf{Q}\mathbf{R}</math>
, where <math display=inline>\mathbf{Q}</math> is unitary and <math display=inline>\mathbf{R}</math> is upper triangular. Inserting the decomposition into the original equality yields <math display=inline>A = \mathbf{B} \mathbf{B}^* = (\mathbf{QR})^*\mathbf{QR} = \mathbf{R}^*\mathbf{Q}^*\mathbf{QR} = \mathbf{R}^*\mathbf{R}</math>. Setting <math display=inline>\mathbf{L} = \mathbf{R}^*</math> completes the proof.