Many techniques in linear algebra make use of properties of inner product spaces and orthogonality. However, when we are working with a finite field such as ${\mathbb{F}}_{q^2}$, where q is a prime power, we cannot define an inner product space since there is no way to define positivity in the finite fields. We can still define a bilinear form $〈,〉:{\mathbb{F}}_{q^2}\to {\mathbb{F}}_{q^2}$ and recover many, but not all, of the techniques and results that have been used over the real and complex numbers. Using the proposed bilinear form, we define two vectors x and y to be orthogonal if $〈x,y〉=0$. We discuss the consequences of defining orthogonality in this manner, explore the existence of self-orthogonal and orthogonal bases, and establish properties of normal and Hermitian matrices.

线性代数中的许多技术都利用了内积空间和正交性的特性。但是，当我们处理有限域时，例如${\mathbb{F}}_{q^{\mathrm{2个}}}$，其中q是素数幂，我们无法定义内积空间，因为无法在有限域中定义正性。我们仍然可以定义双线性形式$〈,〉:{\mathbb{F}}_{q^{\mathrm{2个}}}\to {\mathbb{F}}_{q^{\mathrm{2个}}}$并恢复许多（但不是全部）已用于实数和复数的技术和结果。使用建议的双线性形式，我们将两个向量x和y定义为正交的，如果$〈X,是〉=0$. 我们讨论了以这种方式定义正交性的后果，探讨了自正交和正交基的存在性，并建立了正规矩阵和厄米特矩阵的性质。