Proving 2-Norm of A: Understanding the Relationship Between u and v

[GridironCPJ]

I don't understand how you're supposed to prove this:

Let A=uvT (vT = v transpose) where u is in R^M and v is in R^N. Prove 2-norm of A = 2 norm of v * 2 norm of u.

I'm not sure if I'm supposed to look at v and u as vectors or what. If they are just vectors, this does not make any sense. I'm assuming the only way this is even possible is if v is a collection of m different vectors each of length n, which would just make v a matrix. Am I missing something here?

 

[AlephZero]

It does make sense if u and v are vectors. The product
$$\begin{bmatrix} u_1 \\ u_2 \\ \cdots \\ u_m \end{bmatrix}
\begin{bmatrix} v_1 & v_2 & \cdots & u_n \end{bmatrix}$$ is an m-by-n matrix.

Since there is nothing special about u and v, the only thing you have to work with here is the definition of the 2-norm. Start by multiplying out the matrix, and writing down the values of the three 2-norms.

 

[GridironCPJ]

It does make sense if u and v are vectors. The product
$$\begin{bmatrix} u_1 \\ u_2 \\ \cdots \\ u_m \end{bmatrix}
\begin{bmatrix} v_1 & v_2 & \cdots & u_n \end{bmatrix}$$ is an m-by-n matrix.

Since there is nothing special about u and v, the only thing you have to work with here is the definition of the 2-norm. Start by multiplying out the matrix, and writing down the values of the three 2-norms.

I see what you mean, I was looking at u as a row vector rather than a column vector, which makes multiplication impossible. I multiplied out the matrix, but I'm stuck from here. I can't really use the 2-norm of a matrix since I don't know any of the eigenvalues. What is your tip from here? I didn't quite know what you meant by the "three 2-norms."

 

[AlephZero]

Ahh ... there are too many different definitions of "norm"!

I took the question as meaning the 2-norm of a matrix is
$$\sqrt{\sum_i \sum_j |a_{ij}|^2 }$$
and the 2-norm of a vector is
$$\sqrt{\sum_i |a_{i}|^2 }$$
(i.e. the Frobenius norm). By "the three 2-norms" I just meant the 2-norms of the two vector, and the matrix.

 

[GridironCPJ]

Ahh ... there are too many different definitions of "norm"!

I took the question as meaning the 2-norm of a matrix is
$$\sqrt{\sum_i \sum_j |a_{ij}|^2 }$$
and the 2-norm of a vector is
$$\sqrt{\sum_i |a_{i}|^2 }$$
(i.e. the Frobenius norm). By "the three 2-norms" I just meant the 2-norms of the two vector, and the matrix.

The 2-norm of a matrix is different from the Frobenius norm of a matrix and the proof involves the 2-norm of A, which is a matrix. If the proof were to show equality of the Frobenius norm and the product of the two vector norms, this would make more sense. However, it's for the 2-norm of A.

 

[Office_Shredder]

I can't really use the 2-norm of a matrix since I don't know any of the eigenvalues."

You have a rank 1 matrix and you don't know any of the eigenvalues?

 

[AlephZero]

The 2-norm of a matrix is different from the Frobenius norm of a matrix and the proof involves the 2-norm of A, which is a matrix.

I can think of several definitions of a norm that could be reasonably be called "the 2-norm of a matrix". I can only say what I think the question means, because I don't know what terminology your course is using.