正定矩阵和半正定矩阵的性质及应用数学和应用数学专业论文设计VIP专享VIP免费

摘要本文主要针对正定矩阵和半正定矩阵进行讨论，归纳和总结了正定矩阵和半正定矩阵的性质，通过实例介绍了正定矩阵（半正定矩阵)的判别方法诸如：定义法、主子式法、特征值法等，并且给出了它们在不等式的证明问题中以及多元函数极值问题中的一些应用.关键词：正定矩阵；半正定矩阵；二次型；主子式；特征值ABSTRACTThispapermainlydiscussespositivedefinitematricesandpositivesemi-definitematrix,thepropertiesofpositivedefinitematrixandsemi-positivedefinitematrixaresummarized.Throughexamples,thejudgmentmethodsofpositivedefinitematrixandsemi-positivedefinitematrixareintroduced,suchminormethod,mastertypemethod,eigenvaluemethod,etc.Someapplicationsofpositivedefinitematricesandsemi-positivedefinitematrixintheproofofinequalityextremevalueproblemsofmultivariatefunctionsaregiven.Keywords：positivedefinitematrix;positivesemi-definitematrix;quadraticform;principalminordeterminant;characteristicvalue目录.........................................1..........................................................................................................1..............................................................2...........................................................................2.........................................................................6......................................................................8...................................................................................8.............................................................................13...........................................15............................................................................................................15........................................................................................................15....................................................................................18.......................................................20................................................................................20....................................................................21.....................................................................................................25.............................................................................................................26第1章正定矩阵和半正定矩阵的定义及性质1.1.相关概念定义1[1]设aij(i=1，2，⋯，n，i≤j)都是实常数，则关于n个实变量x1，x2，⋯，xn的二次齐次多项式函数f(x1，x2，⋯，xn)=a11x12+a22x22+⋯+annxn2+2a12x1x2+2a12x1x3+⋯+2an−1，nxn−1xn，称为n元实二次型.[9]定义2[1]实二次型f(x1，x2，⋯，xn)为正定的，如果对于一组不全为零的实数c1，c2，⋯，cn都有f(c1，c2，⋯，cn)>0，如果都有f(c1，c2，⋯，cn)<0，那么称f(x1，x2，⋯，xn)为负定的.如果都有f(c1，c2，⋯，cn)≥0，那么称f(x1，x2，⋯，xn)为半正定的.如果都有f(c1，c2，⋯，cn)≤0,那么称f(x1，x2，⋯，xn)为半负定的.如果二次型既不是半正定又不是半负定，那么称为不定的.[1]定义3[1]若实数域上的n元二次型f(x1，x2，⋯，xn)=∑i=1n∑j=1naijxixj=XTAX是正定（半正定）二次型，则A被称为正定（半正定）矩阵，其中A=(a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮an1an2⋯ann)，X=(x1x2⋮xn)定义4[1]子式|a11a12⋯a1ia21a22⋯a2i⋮⋮⋱⋮ai1ai2⋯aii|，，，，，，，，，0)00()(11121kikjkjiijkkccfccacccf称为矩阵A=(aij)的i阶顺序主子式i=1，2，⋯，n.1.2.正定矩阵和半正定矩阵的等价命题1.2.1.正定矩阵的等价命题定理1[9]A是n阶实对称矩阵，则下列叙述等价：(1)A是正定矩阵.(2)A的所有顺序主子...