任意一个正方矩阵A的二次型xHAx是一实标量。

A . 0 B . 1 C . 2 D . 不存在

A . 1 B . 3 C . 4 D . 5

A .https://assets.asklib.com/images/image2/2017051111260556257.png B .https://assets.asklib.com/images/image2/2017051111261532828.png C .https://assets.asklib.com/images/image2/201705111126258375.png D .https://assets.asklib.com/images/image2/2017051111263281441.png

设f=x^TA x是一个实二次型， 若有实n维向量<img src='https://img2.soutiyun.com/ask/2020-12-29/978125347407201.png' />证明：必有实n维向量<img src='https://img2.soutiyun.com/ask/2020-12-29/97812535927688.png' />

已知二次型<img src='https://img2.soutiyun.com/ask/2020-12-29/978128441590178.png' />经过正交变换化为标准形<img src='https://img2.soutiyun.com/ask/2020-12-29/978128457965115.png' />求参数a、b及所用的正交变换矩阵。

若二阶实对称矩阵A与矩阵<img src='https://img2.soutiyun.com/ask/2020-12-21/977398837755134.png' />合同，则二次型X^TAX的标准形是________

A.<img src='https://img2.soutiyun.com/ask/2020-08-20/966787292961377.png' /> B.<img src='https://img2.soutiyun.com/ask/2020-08-20/966787303034953.png' /> C.<img src='https://img2.soutiyun.com/ask/2020-08-20/96678731188946.png' /> D.<img src='https://img2.soutiyun.com/ask/2020-08-20/966787321664019.png' />

如果实对称矩阵A与矩阵<img src='https://img2.soutiyun.com/ask/2020-08-20/966787469346466.png' />合同，则二次型x^TAx的规范形为(). A.<img src='https://img2.soutiyun.com/ask/2020-08-20/966787498718146.png' /> B.<img src='https://img2.soutiyun.com/ask/2020-08-20/966787513013964.png' /> C.<img src='https://img2.soutiyun.com/ask/2020-08-20/966787525690689.png' /> D.<img src='https://img2.soutiyun.com/ask/2020-08-20/966787536172288.png' />

A．A.(2 4 0 0 5 − 8 0 0 5 ) B．B.(2 4 0 0 5 − 4 0 − 4 5 ) C．C.(2 2 0 2 5 − 4 0 − 4 5 ) D．D.(2 4 0 4 5 − 4 0 − 4 5 )

二次型<img src='https://img2.soutiyun.com/ask/2020-08-17/96652058498053.png' />程的矩阵是() <img src='https://img2.soutiyun.com/ask/2020-08-17/96652059653319.png' />

设二次型<img src='https://img2.soutiyun.com/ask/2020-10-26/972559849328486.jpg' />经正交变换x=Qy，化成<img src='https://img2.soutiyun.com/ask/2020-10-26/972559860523126.jpg' />试求常数a，b。

设实二次型<img src='https://img2.soutiyun.com/ask/2021-01-06/978797001811459.jpg' /><img src='https://img2.soutiyun.com/ask/2021-01-06/978797017600362.jpg' />，证明：f(x₁，x₂，...，x_n)的秩等于矩阵。 <img src='https://img2.soutiyun.com/ask/2021-01-06/978797059757774.jpg' /> 的秩。

若二次型<img src='https://img2.soutiyun.com/ask/2020-08-20/966787609366474.png' /> 为正定的，则t的取值范围是(). A.(2,+∞) B.(- ∞,2) C.(- 1,1) D.<img src='https://img2.soutiyun.com/ask/2020-08-20/966787621249154.png' />

设A为n阶实对称矩阵，R(A)=n; 二次型 <img src='https://img2.soutiyun.com/ask/2020-12-17/977073420019754.png' /> (1)求二次型f的阵 (2) 二次型<img src='https://img2.soutiyun.com/ask/2020-12-17/977073445368204.png' />的规范形是否相同？说明理由.