In applied mathematics, a DFT matrix is an expression of a discrete Fourier transform (DFT) as a transformation matrix, which can be applied to a signal through matrix multiplication. Definition An N-point DFT is expressed as the multiplication , where is the original input signal, is the N-by-N square DFT matrix, and is the DFT of the signal. The transformation matrix can be defined as , or equivalently: , where is a primitive Nth root of unity in which . We can avoid writing large exponents for using the fact that for any exponent we have the identity This is the Vandermonde matrix for the roots of unity, up to the normalization factor. Note that the normalization factor in front of the sum ( ) and the sign of the exponent in ω are merely conventions, and differ in some treatments. All of the following discussion applies regardless of the convention, with at most minor adjustments. The only important thing is that the forward and inverse transforms have opposite-sign exponents, and that the product of their normalization factors be 1/N. However, the choice here makes the resulting DFT matrix unitary, which is convenient in many circumstances. Fast Fourier transform algorithms utilize the symmetries of the matrix to reduce the time of multiplying a vector by this matrix, from the usual . Similar techniques can be applied for multiplications by matrices such as Hadamard matrix and the Walsh matrix. Examples Two-point The two-point DFT is a simple case, in which the first entry is the DC (sum) and the second entry is the AC (difference). The first row performs the sum, and the second row performs the difference. The factor of is to make the transform unitary (see below). Four-point The four-point clockwise DFT matrix is as follows: where . Eight-point The first non-trivial integer power of two case is for eight points: where (Note that .) The following image depicts the DFT as a matrix multiplication, with elements of the matrix depicted by samples of complex exponentials: The real part (cosine wave) is denoted by a solid line, and the imaginary part (sine wave) by a dashed line. The top row is all ones (scaled by for unitarity), so it "measures" the DC component in the input signal. The next row is eight samples of negative one cycle of a complex exponential, i.e., a signal with a fractional frequency of