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例如，对于
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\boldsymbol{F}_{4}=\left[\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & e^{j\left(\frac{\pi}{2}\right)} & e^{j\left(\frac{\pi}{2}\right) 2} & e^{j\left(\frac{\pi}{2}\right) 3} \\ 1 & e^{j\left(\frac{\pi}{2}\right) 2} & e^{j\left(\frac{\pi}{2}\right) 2 * 2} & e^{j\left(\frac{\pi}{2}\right) 2 * 3} \\ 1 & e^{j\left(\frac{\pi}{2}\right) 3} & e^{j\left(\frac{\pi}{2}\right) 3 * 2} & e^{j\left(\frac{\pi}{2}\right) 3 * 3} \end{array}\right]=\left[\begin{array}{cccc} 1 & 1 & 1 & 1 \\1 & i & i^{2} & i^{3} \\1 & i^{2} & i^{4} & i^{6} \\1 & i^{3} & i^{6} & i^{9}\end{array}\right] =\left[\begin{array}{rrrr}1 & 1 & 1 & 1 \\1 & i & -1 & -i \\1 & -1 & 1 & -1 \\1 & -i & -1 & i\end{array}\right]
F4=⎣
⎡11111ej(2π)ej(2π)2ej(2π)31ej(2π)2ej(2π)2∗2ej(2π)3∗21ej(2π)3ej(2π)2∗3ej(2π)3∗3⎦
⎤=⎣
⎡11111ii2i31i2i4i61i3i6i9⎦
⎤=⎣
⎡11111i−1−i1−11−11−i−1i⎦
⎤各个列向量正交（内积
x
i
ˉ
T
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j
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0
\bar{\boldsymbol x_i}^T\boldsymbol x_j=0
xiˉTxj=0），但列向量的模长平方
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ˉ
T
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4
\bar{\boldsymbol x_i}^T\boldsymbol x_i=4
xiˉTxi=4 因此，可以修正得到一个酉矩阵：
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\frac{1}{2}\boldsymbol{F}_{4}=\frac{1}{2}\left[\begin{array}{rrrr}1 & 1 & 1 & 1 \\1 & i & -1 & -i \\1 & -1 & 1 & -1 \\1 & -i & -1 & i\end{array}\right]
21F4=21⎣
⎡11111i−1−i1−11−11−i−1i⎦
⎤ 酉矩阵
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\frac{1}{2}\boldsymbol{F}_{4}
21F4（复数下的正交矩阵）的逆矩阵：
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\frac{1}{2}\boldsymbol{F}_{4}^H
21F4H（共轭转置即可） 满足
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\frac{1}{4} \boldsymbol{F}_{4}{ }^{H} \boldsymbol{F}_{4}=\boldsymbol{I}
41F4HF4=I 另外也可以验证
F
4
\boldsymbol{F}_{4}
F4的逆矩阵：
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F_{4}^{-1}=\frac{1}{4} {F}_{4}^H=\frac{1}{4}\left[\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & e^{-j\left(\frac{\pi}{2}\right)} & e^{-j\left(\frac{\pi}{2}\right) 2} & e^{-j\left(\frac{\pi}{2}\right) 3} \\ 1 & e^{-j\left(\frac{\pi}{2}\right) 2} & e^{-j\left(\frac{\pi}{2}\right) 2 * 2} & e^{-j\left(\frac{\pi}{2}\right) 2 * 3} \\ 1 & e^{-j\left(\frac{\pi}{2}\right) 3} & e^{-j\left(\frac{\pi}{2}\right) 3 * 2} & e^{-j\left(\frac{\pi}{2}\right) 3 * 3} \end{array}\right]
F4−1=41F4H=41⎣
⎡11111e−j(2π)e−j(2π)2e−j(2π)31e−j(2π)2e−j(2π)2∗2e−j(2π)3∗21e−j(2π)3e−j(2π)2∗3e−j(2π)3∗3⎦
⎤
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