| Topic: |
Science > Physics |
| User: |
"" |
| Date: |
12 Dec 2004 05:10:19 PM |
| Object: |
what is maximal rank of this matrix? |
Let n > 2.
Let v(n) and w(n) be any two n-tuplets of real numbers.
Let v(n)' and w(n)' denote the transposes of these n-tuplets.
Let a, b, and c be any 3 real numbers.
Construct the nxn matrix
M(n) = a * v(n) v(n)' + b * w(n) w(n)' + c * { w(n) v(n)' + v(n)
w(n)' }
What is the maximum rank of the nxn matrix M(n)?
[I know the answer is 2, but what is a clever way to PROVE this?]
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| User: "Ray Koopman" |
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| Title: Re: what is maximal rank of this matrix? |
13 Dec 2004 03:03:25 AM |
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wrote:
Let n > 2.
Let v(n) and w(n) be any two n-tuplets of real numbers.
Let v(n)' and w(n)' denote the transposes of these n-tuplets.
Let a, b, and c be any 3 real numbers.
Construct the nxn matrix
M(n) = a * v(n) v(n)' + b * w(n) w(n)' + c * { w(n) v(n)' + v(n)
w(n)' }
What is the maximum rank of the nxn matrix M(n)?
[I know the answer is 2, but what is a clever way to PROVE this?]
The space spanned by (v,w) is at most 2-dimensional.
Mu = 0 for any vector u that is orthogonal to the space;
i.e., for which u'v = u'w = 0.
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| User: "" |
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| Title: Re: what is maximal rank of this matrix? |
21 Dec 2004 01:00:44 AM |
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In article <1102893019.029973.124370@f14g2000cwb.googlegroups.com>, writes:
Let n > 2.
Let v(n) and w(n) be any two n-tuplets of real numbers.
Let v(n)' and w(n)' denote the transposes of these n-tuplets.
Let a, b, and c be any 3 real numbers.
Construct the nxn matrix
M(n) = a * v(n) v(n)' + b * w(n) w(n)' + c * { w(n) v(n)' + v(n)
w(n)' }
What is the maximum rank of the nxn matrix M(n)?
[I know the answer is 2, but what is a clever way to PROVE this?]
If w is a constant multiple of v then M(n) is just a multiple of a
projection operator on the subspace of v, thus of rank 1. If v and w
are independent then, without loss of generality, you can assume that
they are ortonormal (since, if they're not, you can orthonormalize the
base v,w without changing the form of M(n), only changing the values
of the constants. So, assuming that they're orthonormal, you can
create the projection operator P(v,w) = vv' + ww' and show that
PM = MP = 1
The continuation is trivial.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
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