一类特殊正定矩阵的证明问题

The Proof Problem of a Special Positive Definite Matrix

正定矩阵是一类特殊的矩阵,作为对称矩阵的子类,在数学分析中判别多元函数极值、判断函数单调性等具有广泛的应用。他不仅具备对称矩阵可对角化的性质,而且具有对称矩阵不具备的更高的性质。但是要想运用正定矩阵的这些性质,我们就要首先会判断一个矩阵是正定矩阵。可以看到的是在一些课本中或辅导教材中给出了很多正定矩阵的证明方法。一般都是求解特征值,然后证明其特征值大于零。但在一些比较复杂的题目中往往掺杂大量的中间结论的证明,学者在思考过程中如果忽略其中的隐含结论的证明,就会导致整个题目都做不出来。因此做这种题目往往需要我们有大量的知识储备。在本文中我们将要一起讨论针对一类特殊正定矩阵的证明方法,并给出了一种简单的证明方法。

Positive definite matrix is a special kind of matrix. As a subclass of symmetric matrix, positive definite matrix has a wide range of applications in mathematical analysis, such as discriminating the extreme value of multivariate functions and judging the monotonicity of functions. It possesses not only the diagonalization property of symmetric matrices, but also the higher property that symmetric matrices do not possess. But in order to use these properties of positive definite matrices, we first have to know that a matrix is positive definite. And you can see that there are a lot of proofs for positive definite matrices that are given in some books or in some tutorial books. You usually solve for the eigenvalues, and then you prove that the eigenvalues are greater than zero. However, in some more complex topics, a large number of proofs of intermediate conclusions are often mixed in. If scholars ignore the proofs of implicit conclusions in the process of thinking, they will fail to complete the whole topic. Therefore, to do this kind of problem often requires us to have a large knowledge reserve. Today we are going to discuss the proof for a special class of matrices, and give a simple proof method.