Editing Projection (linear algebra) (section)

==== Singular values ====
<math>I-P</math> is also an oblique projection. The singular values of <math>P</math> and <math>I-P</math> can be computed by an [[orthonormal basis]] of <math>A</math>. Let 
<math>Q_A</math> be an orthonormal basis of <math>A</math> and let <math>Q_A^{\perp}</math> be the [[orthogonal complement]] of <math>Q_A</math>. Denote the singular values of the matrix
<math>Q_A^T A (B^T A)^{-1} B^T Q_A^{\perp} </math> by the positive values <math>\gamma_1 \ge \gamma_2 \ge \ldots \ge \gamma_k </math>. With this, the singular values for <math>P</math> are:<ref>{{Citation | last1 = Brust | first1 = J. J. | last2 = Marcia | first2 = R. F. | last3 = Petra | first3 = C. G. | date = 2020 | title = Computationally Efficient Decompositions of Oblique Projection Matrices | journal = SIAM Journal on Matrix Analysis and Applications | volume = 41 | issue = 2 | pages =  852–870 | doi=10.1137/19M1288115 | osti = 1680061 | s2cid = 219921214 }}</ref> 
<math display="block">\sigma_i = 
	\begin{cases}
		\sqrt{1+\gamma_i^2} & 1 \le i \le k \\
		0 & \text{otherwise}
	\end{cases} 
	</math>
and the singular values for <math>I-P</math> are
<math display="block">\sigma_i = 
\begin{cases}
		\sqrt{1+\gamma_i^2} 	& 1 \le i \le k \\
		1 				& k+1 \le i \le n-k \\
		0 & \text{otherwise}
	\end{cases} 
	</math>
This implies that the largest singular values of <math>P</math> and <math>I-P</math> are equal, and thus that the [[matrix norm]] of the oblique projections are the same. However, the [[condition number]] satisfies the relation <math>\kappa(I-P) = \frac{\sigma_1}{1} \ge \frac{\sigma_1}{\sigma_k} = \kappa(P)</math>, and is therefore not necessarily equal.