# proof question about positive semidefinite matrix(important 4 regression analysis)

• May 30th 2009, 01:16 AM
phoenicks
proof question about positive semidefinite matrix(important 4 regression analysis)
this is a proof I encounter in William Greene econometric analysis appendix section.

If A is a n*k with full column rank and n > k, then (A')(A)is positive definite and (A)(A') is positive semi-definite.

proof given by author is as follows,

By assumption, Ax is not equal to zero. So, x'A'Ax = (Ax)'(Ax) = y'y = summation y^2 > 0

for the latter case, because A has more rows then columns, then there is an X such that A'x = 0, thus we can only have y'y >= 0

What I dont understand is the the bold and underline part.

P/S: this question comes from pg 835, of William Greene Econometric Analysis 5th edition textbook.

• May 30th 2009, 12:42 PM
NonCommAlg
Quote:

Originally Posted by phoenicks
this is a proof I encounter in William Greene econometric analysis appendix section.

If A is a n*k with full column rank and n > k, then (A')(A)is positive definite and (A)(A') is positive semi-definite.

proof given by author is as follows,

By assumption, Ax is not equal to zero. So, x'A'Ax = (Ax)'(Ax) = y'y = summation y^2 > 0

\$\displaystyle A\$ has full column rank means that the columns of \$\displaystyle A\$ are linearly independent. see that if \$\displaystyle v_1, \cdots , v_k\$ are the columns of \$\displaystyle A\$ and \$\displaystyle \bold{x}=[x_1 \ x_2 \cdots \ x_k]^T,\$ then \$\displaystyle A \bold{x}=x_1v_1 + \cdots + x_kv_k.\$

so if \$\displaystyle A \bold{x}=\bold{0},\$ then \$\displaystyle x_1v_1 + \cdots + x_k v_k = 0,\$ and thus \$\displaystyle x_j = 0,\$ for all \$\displaystyle j,\$ because \$\displaystyle v_1, \cdots , v_k\$ are linearly independent. hence the only solution of \$\displaystyle A \bold{x}=\bold{0}\$ is \$\displaystyle \bold{x}=\bold{0}.\$

Quote:

for the latter case, because A has more rows then columns, then there is an X such that A'x = 0, thus we can only have y'y >= 0

What I dont understand is the the bold and underline part.

P/S: this question comes from pg 835, of William Greene Econometric Analysis 5th edition textbook.