Editing QR decomposition (section)

==Connection to a determinant or a product of eigenvalues==
We can use QR decomposition to find the [[determinant]] of a square matrix. Suppose a matrix is decomposed as <math>A = QR</math>. Then we have
<math display='block'>\det A = \det Q \det R.</math>

<math>Q</math> can be chosen such that <math>\det Q = 1</math>. Thus,
<math display='block'>\det A = \det R = \prod_i r_{ii}</math>

where the <math>r_{ii}</math> are the entries on the diagonal of <math>R</math>. Furthermore, because the determinant equals the product of the eigenvalues, we have
<math display='block'>\prod_{i} r_{ii} = \prod_{i} \lambda_{i}</math>

where the <math>\lambda_i</math> are eigenvalues of <math>A</math>.

We can extend the above properties to a non-square complex matrix <math>A</math> by introducing the definition of QR decomposition for non-square complex matrices and replacing eigenvalues with singular values.

Start with a QR decomposition for a non-square matrix ''A'':
: <math>A = Q \begin{bmatrix} R \\ 0 \end{bmatrix}, \qquad Q^\dagger Q = I</math>

where <math>0</math> denotes the zero matrix and <math>Q</math> is a unitary matrix.

From the properties of the [[singular value decomposition]] (SVD) and the determinant of a matrix, we have
:<math>\Big|\prod_i r_{ii}\Big| = \prod_i\sigma_{i},</math>

where the <math>\sigma_i</math> are the singular values of {{nowrap|<math>A</math>.}}

Note that the singular values of <math>A</math> and <math>R</math> are identical, although their complex eigenvalues may be different. However, if ''A'' is square, then
:<math>{\prod_i \sigma_i} = \Big|\prod_i \lambda_i\Big|.</math>

It follows that the QR decomposition can be used to efficiently calculate the product of the eigenvalues or singular values of a matrix.