Form of symmetric matrix of rank one

Also, since C has rank one, it has only one non-zero eigenvalue. This means there exists a vector w such that C=aww^T. Now, we just need to show that w has norm one and a is a scalar.
  • #1
ianchenmu
10
0

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
ianchenmu said:

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?
 

Related to Form of symmetric matrix of rank one

1. What is a symmetric matrix of rank one?

A symmetric matrix of rank one is a square matrix where all the elements are symmetric about the main diagonal, and the rank of the matrix is one. This means that the matrix has only one linearly independent row or column.

2. How can a matrix be symmetric and have a rank of one?

Since a symmetric matrix of rank one only has one linearly independent row or column, the remaining rows or columns can be expressed as linear combinations of this independent row or column. This results in a matrix that is symmetric, but with a rank of one.

3. What is the significance of a symmetric matrix of rank one?

A symmetric matrix of rank one is often used in data analysis and statistics to represent relationships between variables. It can also be used in linear algebra to simplify calculations and solve systems of equations.

4. How can you determine if a matrix is symmetric and of rank one?

To determine if a matrix is symmetric, you can check if it is equal to its transpose. If all the elements are symmetric about the main diagonal, then the matrix is symmetric. To determine the rank, you can use row or column operations to reduce the matrix to its echelon form and count the number of linearly independent rows or columns.

5. Can a symmetric matrix of rank one be diagonalizable?

Yes, a symmetric matrix of rank one can be diagonalizable. Since the matrix is already symmetric, it is similar to its diagonal form. This means that the matrix can be expressed as a product of a diagonal matrix and its transpose, which are both symmetric.

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