Symmetric and alternate matrices in an arbitrary field. I
Symmetric and alternate matrices in an arbitrary field. I
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任意域中的对称矩阵和交替矩阵。
DOI：
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发表时间：
1938
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通讯作者：
A. Albert
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文献类型：
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作者：
A. Albert
在不改变证明的情况下，对称矩阵和交错矩阵的经典处理的基本定理可以被证明为适用于元素在任何特征域中而不是两个的矩阵。证明在特征两种情况下失败，结果不成立，因为对称矩阵和交替矩阵的概念在这里重合。但有可能获得统一的治疗。在这里，我们将通过在交替矩阵的定义中添加一个条件来提供这一点，该条件除了特征二的域之外是多余的。经典结果的证明将通过添加两个必要的新论点来完成。实对称矩阵的定定性定理在一般领域中没有类似的结果。它们都是基于任意两个非负实数的和是非负的这一性质。这等价于对于每个实数a和b，我们对一个实数c有a2+b2=c2，但在任何特征为2的域中，a2？b2=(a+b)2，我们将利用这一事实得到特征为2的任意域的完全类比，这是关于实对称矩阵的确定性的通常定理。二次型可以与对称矩阵联系在一起，它们的等价性问题等价于相应矩阵的同余问题。这是正确的，除非参照场具有特征二，其中没有给予矩阵处理。我们将在这种情况下将二次型与某种类型的非对称矩阵联系起来，并将使用我们关于交替矩阵同余的结果来获得二次型问题的矩阵处理。本文将证明关于具有复元素的对称或交错矩阵对的经典定理对于元素在任何特征不为二的代数闭域上的矩阵是成立的。这意味着任何两个对称(或交替)矩阵是正交等价的，当且仅当它们相似。但对于特征二的领域，证明是失败的。
The elementary theorems of the classical treatment of symmetric and alternate matrices may be shown, without change in the proofs, to hold for matrices whose elements are in any field of characteristic not two. The proofs fail in the characteristic two case and the results cannot hold since here the concepts of symmetric and alternate matrices coincide. But it is possible to obtain a unified treatment. We shall provide this here by adding a condition to the definition of alternate matrices which is redundant except for fields of characteristic two. The proofs of the classical results will then be completed by the addition of two necessary new arguments. The theorems on the definiteness of real symmetric matrices have had no analogues for general fields. They have been based on the property that the sum of any two non-negative real numbers is non-negative. This is equivalent to the property that for every real a and b we have a 2+b2 =c2 for a real c. But a2?b2 = (a+b)2 in any field of characteristic two and we shall use this fact to obtain complete analogues for arbitrary fields of characteristic two of the usual theorems on the definiteness of real symmetric matrices. Quadratics forms may be associated with symmetric matrices and the problem of their equivalence is equivalent to the problem of the congruence of the corresponding matrices. This is true except when the field of reference has characteristic two where no matric treatment has been given. We shall associate quadratic forms in this case with a certain type of non-symmetric matrix and shall use our results on the congruence of alternate matrices to obtain a matrix treatment of the quadratic form problem. The classical theoremst on pairs of symmetric or alternate matrices with complex elements will be shown here to be true for matrices with elements in any algebraically closed field whose characteristic is not two. This will be seen to imply that any two symmetric (or alternate) matrices are orthogonally equivalent if and only if they are similar. But the proof fails for fields of characteristic two.