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Invertible Matrix

  • Thread starter roam
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  • #1
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Homework Statement



If A is a an mxn matrix and its column vectors are linearly independent.

Prove that the matrix AtA is nonsingular. Hint: Use the fact that it is sufficient to show that null(AtA) = {0}

Homework Equations





The Attempt at a Solution



I'm new to this topic & I don't understand the hint given and how exactly to use it to prove the question...

I know that in order for a matrix to be nonsingular/invertible it has to be squre (m=n) and when you multiply the a matrix by its transpose, the resulting matrix would be square.
I'm also thinking about the properties of fundamental spaces of matrices that: row(A) = null(A) and col(A)=null(At) (therefore null(AtA) = row(A).col(A)?)

Any help would be much appreciated :)

Cheers.
 

Answers and Replies

  • #2
Dick
Science Advisor
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If the columns of A are linearly independent, doesn't that mean null(A)={0}? Now suppose A^(T)A were singular and think about x^(T)A^(T)*Ax, where x is a column vector in R^n.
 

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