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2014年矩阵论试题A

2014年矩阵论试题A
2014年矩阵论试题A

长 春 理 工 大 学

研 究 生 期 末 考 试 试 题 科目名称: 矩 阵 论 命题人:姜志侠 适用专业: 理 工 科 审核人: 开课学期:2013 ——2014 学年第 一 学期 □开卷 √闭卷

一、(10分)F 为数域,对于线性空间22?F 中任意矩阵???

? ??=d c b a A ,规则σ,τ分别为???

? ??=???? ??=c a A c b a A )(,0)(τσ,问σ,τ是否为22?F 上的变换,如果是,证明该变换为线性变换,并求该变换在基???? ??=000111E ,???? ??=001012E ,???? ??=010021E ,???

? ??=100022E 下的矩阵.

二、(10分) 已知正规矩阵???

? ??-=1111A ,求酉矩阵U ,使得AU U H 为对角形矩阵。三、(10分) 用Schmidt 正交化方法求矩阵????

? ??=101011110A 的QR 分解.

四、(10分) 设矩阵??????

? ?

?-=2000120010201012A ,求A 的行列式因子,不变因子,初等因子组, Jordan 标准形。 五、(10分) 求可对角化矩阵460350361A ?? ?=-- ? ?--??

的谱分解式.

六、(10分) 在线性空间n m C ?中,对任意矩阵n m ij a A ?=)(,定义函数ij j

i a mn A ,max ?=,证明此函数是矩阵范数。

七、(10分) 已知函数矩阵

????

??????=32010cos sin )(x x e x x

x x A x ,

其中0≠x ,试求)(lim 0x A x →,dx x dA )(,2

2)(dx x A d ,dx x dA )(. 八、(10分)已知矩阵??

????--=1244916A ,写出矩阵函数)(A f 的Lagrange-Sylvester 内插多项式表示,并计算A πcos .

.

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