Connection To A Determinant or A Product of Eigenvalues
We can use QR decomposition to find the absolute value of the determinant of a square matrix. Suppose a matrix is decomposed as . Then we have
Since Q is unitary, . Thus,
where are the entries on the diagonal of R.
Furthermore, because the determinant equals the product of the eigenvalues, we have
where are eigenvalues of .
We can extend the above properties to non-square complex matrix by introducing the definition of QR-decomposition for non-square complex matrix and replacing eigenvalues with singular values.
Suppose a QR decomposition for a non-square matrix A:
where is a zero matrix and is an unitary matrix.
From the properties of SVD and determinant of matrix, we have
where are singular values of .
Note that the singular values of and are identical, although the complex eigenvalues of them may be different. However, if A is square, it holds that
In conclusion, QR decomposition can be used efficiently to calculate a product of eigenvalues or singular values of matrix.
Read more about this topic: QR Decomposition
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