# Orthonormal and Unitary Transformations

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## Abstract

In Gaussian elimination and LU factorization we solve a linear system by transforming it to triangular form. These are not the only kind of transformations that can be used for such a task. Matrices with orthonormal columns, called unitary matrices can be used to reduce a square matrix to upper triangular form and more generally a rectangular matrix to upper triangular (also called upper trapezoidal) form. This lead to a decomposition of a rectangular matrix known as a **QR decomposition** and a reduced form which we refer to as a **QR factorization**. The QR decomposition and factorization will be used in later chapters to solve least squares- and eigenvalue problems.

## References

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