求教一道线性代数的题目 [推广有奖]

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tianyu0401 发表于 2009-9-18 23:21:49 |AI写论文

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Suppose X is an n × K matrix with full column rank of K. Show that
X的转秩乘以X is nonsigular (invertible).

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关键词：线性代数 Suppose matrix column colum 题目线性代数求教

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sherman_shi 发表于2楼查看完整内容

Suppose A is an n × K matrix with full column rank of K. Show that A'A (K×K) is nonsigular (invertible), which means Rank(A'A)=Rank(A)=K. Prove equality of their null spaces. Null space of the A'A matrix is given by vectors x for which A'Ax=0. If this condition is fulfilled, also holds 0=x'A'Ax=0, which means Ax=0. On the other hand, if Ax=0, then A'Ax=0. So, their null spaces are same ...

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sherman_shi 发表于 2009-9-19 00:08:07

Suppose A is an n × K matrix with full column rank of K. Show that
A'A (K×K) is nonsigular (invertible), which means Rank(A'A)=Rank(A)=K.

Prove equality of their null spaces. Null space of the A'A matrix is given by vectors x for which A'Ax=0. If this condition is fulfilled, also holds 0=x'A'Ax=0, which means Ax=0.

On the other hand, if Ax=0, then A'Ax=0.

So, their null spaces are same, and therefore Rank(A'A)=Rank(A)=K.