Hello All,

Can anyone explain what is the physical meaning or maybe the
statistical meaning
of the matrix norm of a covariance matrix?

To be more specific:
Let X be a Gaussian random vector with A being its covariance matrix.

and M = norm(A);

where norm is the norm defined by:

norm(A) = max ||Ax||/||x||  (on all vectors x)

what is the meaning of M?

Even if I try to think on M as the maximal eigenvalue of A'A ( ' -
stands for transposed )
still, I cant understand the meaning of this.

On Apr 9, 11:19 am, miki <miki.li...@gmail.com
wrote:
Hello All,

Can anyone explain what is the physical meaning or
maybe the
statistical meaning
of the matrix norm of a covariance matrix?

To be more specific:
Let X be a Gaussian random vector with A being its
covariance matrix.

and M = norm(A);

where norm is the norm defined by:

norm(A) = max ||Ax||/||x|| (on all vectors x)

what is the meaning of M?

Even if I try to think on M as the maximal
eigenvalue of A'A ( ' -
stands for transposed )
still, I cant understand the meaning of this.

Since the covariance matrix is symmetric, think of it
as the
maximal eigenvalue of A (not A'A) and therefore, the
maximum
magnification possible via multiplication by A.

Hope this helps.

Greg

On Apr 9, 11:19 am, miki <miki.li...@gmail.com
wrote:
Hello All,

Can anyone explain what is the physical meaning or
maybe the
statistical meaning
of the matrix norm of a covariance matrix?

To be more specific:
Let X be a Gaussian random vector with A being its
covariance matrix.

and M = norm(A);

where norm is the norm defined by:

norm(A) = max ||Ax||/||x||  (on all vectors x)

what is the meaning of M?

Even if I try to think on M as the maximal
eigenvalue of A'A ( ' -
stands for transposed )
still, I cant understand the meaning of this.

Since the covariance matrix is symmetric, think of it
as the
maximal eigenvalue of A (not A'A) and therefore, the
maximum
magnification possible via multiplication by A.

Hope this helps.

Greg

Greg's interpretation is correct.  Another interpretation is that this norm is the variance of the first principal component; that is, the largest variance of all linear combinations Ax such that x'x = 1.

Here's the derivation that shows that this norm is the largest eigenvalue of A.

norm(A) = max ||Ax||/||x|| = max sqrt(x'AA'x/x'x) > sqrt max(x'(A^2)x/x'x) = sqrt(ch_max(A^2)) = sqrt(ch_max(A)^2) = ch_max(A).

Jack- Hide quoted text -

- Show quoted text -

On Apr 9, 10:41 pm, Jack Tomsky <jtom...@ix.netcom.com> wrote:
On Apr 9, 11:19 am, miki <miki.li...@gmail.com
wrote:
Hello All,
Can anyone explain what is the physical meaning or
maybe the
statistical meaning
of the matrix norm of a covariance matrix?
To be more specific:
Let X be a Gaussian random vector with A being its
covariance matrix.
and M = norm(A);
where norm is the norm defined by:
norm(A) = max ||Ax||/||x|| (on all vectors x)
what is the meaning of M?
Even if I try to think on M as the maximal
eigenvalue of A'A ( ' -
stands for transposed )
still, I cant understand the meaning of this.
Since the covariance matrix is symmetric, think of it
as the
maximal eigenvalue of A (not A'A) and therefore, the
maximum
magnification possible via multiplication by A.
Hope this helps.
Greg
Greg's interpretation is correct. Another interpretation is that this norm is the variance of the first principal component; that is, the largest variance of all linear combinations Ax such that x'x = 1.

Here's the derivation that shows that this norm is the largest eigenvalue of A.

norm(A) = max ||Ax||/||x|| = max sqrt(x'AA'x/x'x) =
sqrt max(x'(A^2)x/x'x) = sqrt(ch_max(A^2)) = sqrt(ch_max(A)^2) = ch_max(A).

Jack- Hide quoted text -

- Show quoted text -

Well,
Ax is a vector so what did you mean by its variance. (Norm is a scalar
value)
Did you mean that norm(A) is the maximum variance of ||Ax|| amongs all
x such that x'x = 1?
If so, why?

Thanks,
Miki