In this chapter we look at the QR decomposition of a matrix. Any matrix \({\boldsymbol {A}}\) can be factored into the product \(\boldsymbol {A} = \boldsymbol {QR}\) , where \({\boldsymbol {Q}}\) is orthonormal and \({\boldsymbol {R}}\) is upper triangular. The QR decomposition is very useful in that it provides an easily computable orthonormal basis for \(R(\boldsymbol {A})\) . In this way, it is like a “poor-man’s” SVD. We discuss several computational techniques for the QR decomposition—these include the classical Gram–Schmidt method, which is the most straightforward to understand but comes at the cost of potential for numerical error. We also discuss the modified Gram–Schmidt method, which is similar to the classical approach but resolves the issue of numerical error. We also discuss the Householder method, which is the most commonly used method since it is computationally fast and numerically stable. We extend the treatment of the QR decomposition to the rank-deficient case. As we see later, the QR decomposition provides an efficient means of solving both the rank-deficient and the full-rank least squares problem. We close the chapter by introducing the QR method for computation of the eigendecomposition.