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Science Forum Index » Math - Numerical Analysis Forum » Symmetrizing triangular submatrices
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Message |
| Tobias Burnus |
 Posted: Wed Mar 07, 2007 10:32 am |
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Hi all,
I have a real, Hermitian matrix
( H11 H12 )
( H21 H22 ) = H,
where H11 and H22 are 5x5 matrices and the H12 and H21 are triangular
matrices.
I would now like to find a unitary transformation, which makes the
submatrix H12 as symmetric as possible.
Any idea how to achieve this?
Tobias
Example matrix (first column = column index)
------------------------------------------------------------------------------
1 1.00 0.00 0.00 0.00 0.00 | 1.92 0.00 0.00 0.00 0.00
2 0.00 1.00 0.00 0.00 0.00 | 0.00 2.59 0.00 0.00 0.00
3 0.00 0.00 1.00 0.00 0.00 | 1.35 0.00 2.21 0.00 0.00
4 0.00 0.00 0.00 1.00 0.00 | 0.00 1.00 0.00 1.64 0.00
5 0.00 0.00 0.00 0.00 1.00 | 0.00 0.00 0.00 0.00 1.64
------------------------------------------------------------------------------
6 1.92 0.00 1.35 0.00 0.00 | 1.15-0.02-0.61 0.06-0.03
7 0.00 2.59 0.00 1.00 0.00 | -0.02 0.40-0.01-0.77-1.32
8 0.00 0.00 2.21 0.00 0.00 | -0.61-0.01-0.91 0.02 0.00
9 0.00 0.00 0.00 1.64 0.00 | 0.06-0.77 0.02-0.03-0.46
10 0.00 0.00 0.00 0.00 1.64 | -0.03-1.32 0.00-0.46-0.05
------------------------------------------------------------------------------ |
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| Peter Spellucci |
| Posted: Wed Mar 07, 2007 10:32 am |
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In article <1173277971.075499.27700@n33g2000cwc.googlegroups.com>,
"Tobias Burnus" <burnus@net-b.de> writes:
Quote: Hi all,
I have a real, Hermitian matrix
( H11 H12 )
( H21 H22 ) = H,
where H11 and H22 are 5x5 matrices and the H12 and H21 are triangular
matrices.
I would now like to find a unitary transformation, which makes the
submatrix H12 as symmetric as possible.
Any idea how to achieve this?
Tobias
Example matrix (first column = column index)
------------------------------------------------------------------------------
1 1.00 0.00 0.00 0.00 0.00 | 1.92 0.00 0.00 0.00 0.00
2 0.00 1.00 0.00 0.00 0.00 | 0.00 2.59 0.00 0.00 0.00
3 0.00 0.00 1.00 0.00 0.00 | 1.35 0.00 2.21 0.00 0.00
4 0.00 0.00 0.00 1.00 0.00 | 0.00 1.00 0.00 1.64 0.00
5 0.00 0.00 0.00 0.00 1.00 | 0.00 0.00 0.00 0.00 1.64
------------------------------------------------------------------------------
6 1.92 0.00 1.35 0.00 0.00 | 1.15-0.02-0.61 0.06-0.03
7 0.00 2.59 0.00 1.00 0.00 | -0.02 0.40-0.01-0.77-1.32
8 0.00 0.00 2.21 0.00 0.00 | -0.61-0.01-0.91 0.02 0.00
9 0.00 0.00 0.00 1.64 0.00 | 0.06-0.77 0.02-0.03-0.46
10 0.00 0.00 0.00 0.00 1.64 | -0.03-1.32 0.00-0.46-0.05
------------------------------------------------------------------------------
if you diagonalize the whole thing, then finally H12=H21=O perfectly symmetric
but I guess you wanted something more direct?
hth
peter |
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| kunzmilan |
| Posted: Thu Mar 08, 2007 4:54 am |
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On Mar 7, 6:10 pm, spellu...@fb04373.mathematik.tu-darmstadt.de (Peter
Spellucci) wrote:
Quote: In article <1173277971.075499.27...@n33g2000cwc.googlegroups.com>,
"Tobias Burnus" <bur...@net-b.de> writes:
Hi all,
I have a real, Hermitian matrix
( H11 H12 )
( H21 H22 ) = H,
where H11 and H22 are 5x5 matrices and the H12 and H21 are triangular
matrices.
I would now like to find a unitary transformation, which makes the
submatrix H12 as symmetric as possible.
Any idea how to achieve this?
Tobias
Example matrix (first column = column index)
------------------------------------------------------------------------------
1 1.00 0.00 0.00 0.00 0.00 | 1.92 0.00 0.00 0.00 0.00
2 0.00 1.00 0.00 0.00 0.00 | 0.00 2.59 0.00 0.00 0.00
3 0.00 0.00 1.00 0.00 0.00 | 1.35 0.00 2.21 0.00 0.00
4 0.00 0.00 0.00 1.00 0.00 | 0.00 1.00 0.00 1.64 0.00
5 0.00 0.00 0.00 0.00 1.00 | 0.00 0.00 0.00 0.00 1.64
------------------------------------------------------------------------------
6 1.92 0.00 1.35 0.00 0.00 | 1.15-0.02-0.61 0.06-0.03
7 0.00 2.59 0.00 1.00 0.00 | -0.02 0.40-0.01-0.77-1.32
8 0.00 0.00 2.21 0.00 0.00 | -0.61-0.01-0.91 0.02 0.00
9 0.00 0.00 0.00 1.64 0.00 | 0.06-0.77 0.02-0.03-0.46
10 0.00 0.00 0.00 0.00 1.64 | -0.03-1.32 0.00-0.46-0.05
------------------------------------------------------------------------------
if you diagonalize the whole thing, then finally H12=H21=O perfectly symmetric
but I guess you wanted something more direct?
hth
peter
Transform as
H22 H21
H12 H11,
and then diagonalize H12.
kunzmilan |
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| Tobias Burnus |
| Posted: Tue Mar 13, 2007 8:54 am |
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On Mar 8, 9:54 am, "kunzmilan" <kunzmi...@atlas.cz> wrote:
Quote: On Mar 7, 6:10 pm, spellu...@fb04373.mathematik.tu-darmstadt.de (Peter
Spellucci) wrote:
I have a real, Hermitian matrix
( H11 H12 )
( H21 H22 ) = H,
where H11 and H22 are 5x5 matrices and the H12 and H21 are triangular
matrices.
I would now like to find a unitary transformation, which makes the
submatrix H12 as symmetric as possible.
Transform as
H22 H21
H12 H11,
Ok, I have now
( H22 H21)
( H12 H11) = A,
where the H11, H12, H21, H22 matrices are unchanged from above. Or did
I misunderstood something (such as an transposed or ...)?
Quote: and then diagonalize H12.
If I use the unmodified matrix H12, why should I construct A first?
Tobias |
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