# When is the Frobenius norm of a matrix equal to the 2-norm of a matrix?

What conditions most be true for these two norms to be equal? Or are they always equal?

What conditions most be true for these two norms to be equal? Or are they always equal?

I'm far from being a specialist in this, but it seems to me that "Frobenius norm of a matrix" is just the name given to the 2-norm...

Don

Well, in the applied linear algebra course I'm taking currently, the Frobenius norm of a matrix A is defined as the square root of the trace of A'A and the 2-norm is defined as the square root of the largest eigenvalue of A'A. I'm just not sure if they're always the same.

The Frobenius and 2-norm of a matrix coincide if and only if the matrix has rank 1 (i.e. if and only if the matrix can be represented as A=c r, where r is a row and c is a column).

You can see that from the fact that Frobenius norm is $\left( \sum_k s_k^2\right)^{1/2}$ and the 2-norm is $\max s_k$, where $s_k$ are singular values. So equality happens if and only if there is only one non-zero singular value, which is equivalent to the fact that the rank is 1.

The Frobenius and 2-norm of a matrix coincide if and only if the matrix has rank 1 (i.e. if and only if the matrix can be represented as A=c r, where r is a row and c is a column).

You can see that from the fact that Frobenius norm is $\left( \sum_k s_k^2\right)^{1/2}$ and the 2-norm is $\max s_k$, where $s_k$ are singular values. So equality happens if and only if there is only one non-zero singular value, which is equivalent to the fact that the rank is 1.

Excellent, thank you. The matrix in a proof I'm working on involves a rank 1 matrix, so this equality of the two norms applies perfectly.

AlephZero
Homework Helper
The Frobenius and 2-norm of a matrix coincide if and only if the matrix has rank 1

More generally, ##||A||_2 \le ||A||_F \le \sqrt{r}||A||_2## where r is the rank of A.

More generally, ##||A||_2 \le ||A||_F \le \sqrt{r}||A||_2## where r is the rank of A.

May you shed some light on this? Or quote any possible reference? Thanks

jbunniii
Homework Helper
Gold Member
May you shed some light on this? Or quote any possible reference? Thanks
Assuming you accept Hawkeye18's formulas, namely
$$\|A\|_F = \left( \sum_k s_k^2\right)^{1/2}$$
and
$$\|A\|_2 = \max{s_k}$$
then we have
$$\|A\|_2 = \max{s_k} = \left( (\max{s_k})^2\right)^{1/2} \leq \left( \sum_{k} s_k^2 \right)^{1/2} = \|A\|_F$$

For the second inequality, note that the rank of ##A## is precisely the number of nonzero singular values. Let's sort the singular values so that the nonzero ones all come first. Then for a rank ##r## matrix, we have
$$\|A\|_F = \left( \sum_{k=1}^{r} s_k^2\right) ^{1/2} \leq \left( \sum_{k=1}^{r} (\max s_k)^2 \right)^{1/2} = (r (\max s_k)^2)^{1/2} = \sqrt{r} \|A\|_2$$
Equality holds if and only if the ##r## nonzero singular values are all equal.

hairetikos and jim mcnamara
when matrix A is Singular which means det(A)=0.