正定二次型在《高等代数》考研试题中的应用

Application of Positive Definite Quadratic Form in Postgraduate Entrance Examination Questions of Advanced Algebra

摘要: 正定二次型是《高等代数》中实二次型中的一类特殊形式，也是《高等代数》考研试题中的题型之一，探讨正定二次型在《高等代数》考研试题中的应用有很好的现实意义。文章首先对正定二次型的相关知识进行了简单阐述，然后重点对正定二次型在考研试题中的应用从四个方面进行了探讨。具体从正定二次型在二次型的性质中的应用，正定二次型在矩阵秩中的应用、在矩阵对角化中的应用、在分块矩阵中的应用四方面进行分析，并用近年的考研真题进行了剖析，对数学与应用数学考研学生有一定的应用价值。

Abstract: Positive definite quadratic form is a special type of real quadratic form in Advanced Algebra, and it is also one of the question types in the postgraduate entrance examination questions of Advanced Algebra. It is of great practical significance to discuss the application of positive definite quadratic form in the postgraduate entrance examination of Advanced Algebra. Firstly, the article briefly expounds on the relevant knowledge of the positive definite quadratic form, and then focuses on the application of the positive definite quadratic form in the postgraduate examination questions from four aspects. Specifically, it analyzes the application of positive definite quadratic form in the properties of quadratic form, the application of positive definite quadratic form in matrix rank, the application in matrix diagonalization, and the application in block matrix, and analyzes the real questions of postgraduate entrance examination in recent years, which has certain application value for postgraduate students of mathematics and applied mathematics.