QR Decomposition - Connection To A Determinant or A Product of Eigenvalues

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

{\prod_{i} \sigma_{i}} = \Big|{\prod_{i} \lambda_{i}}\Big|.

In conclusion, QR decomposition can be used efficiently to calculate a product of eigenvalues or singular values of matrix.

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