Fourier transforms form an integral part of most signal processing applications and are therefore a crucial enabler of modern technology. Thus, fast and efficient algorithms to compute the Fourier transform is an area of active research. A class of algorithms called the Fast Fourier Transforms (FFTs) already exist which can compute the Discrete Fourier Transform (DFT) quite efficiently. However, these algorithms use complex arithmetic. This could be a hindrance in hardware that lacks dedicated multiplier units like low power/low performance digital signal processors and embedded microprocessors. Moreover, these algorithms are not easily vectorizable. To avoid these drawbacks, a natively-real DFT computational algorithm is presented in this dissertation. This can compute the DFT with real arithmetic using real DFT eigenvectors. Moreover, this algorithm is entirely based on linear algebra and is not as complicated as the FFT algorithms. This can be implemented in parallel processing hardware since it is easily vectorizable. It has the potential to consume less power on low performance or fixed-point hardware although it needs more number of operations than existing FFT algorithms. If the need arises, this method can be combined with FFT algorithms to reduce the number of operations. This dissertation also presents a few techniques to obtain the real and orthonormal DFT eigenvectors from the DFT projection matrices. For each of these techniques, the DFT computation error and the number of operations required are also discussed. A 2-dimensional DFT computation algorithm is also presented.