Form of symmetric matrix of rank one

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Homework Statement



The question is:


Let [itex]C[/itex] be a symmetric matrix of rank one. Prove that [itex]C[/itex] must have the form [itex]C=aww^T[/itex], where [itex]a[/itex] is a scalar and [itex]w[/itex] is a vector of norm one.




Homework Equations


n/a


The Attempt at a Solution


I think we can easily prove that if [itex]C[/itex] has the form [itex]C=aww^T[/itex], then [itex]C[/itex] is symmetric and of rank one. But what about the opposite direction...that is what we need to prove. How to prove this?
 

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  • #2
Dick
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Homework Statement



The question is:


Let [itex]C[/itex] be a symmetric matrix of rank one. Prove that [itex]C[/itex] must have the form [itex]C=aww^T[/itex], where [itex]a[/itex] is a scalar and [itex]w[/itex] is a vector of norm one.




Homework Equations


n/a


The Attempt at a Solution


I think we can easily prove that if [itex]C[/itex] has the form [itex]C=aww^T[/itex], then [itex]C[/itex] is symmetric and of rank one. But what about the opposite direction...that is what we need to prove. How to prove this?

Do you know that if C is symmetric, it can be diagonalized?
 

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