Re: schur decomposition and mathematica

• To: mathgroup at smc.vnet.net
• Subject: [mg66951] Re: schur decomposition and mathematica
• From: bghiggins at ucdavis.edu
• Date: Mon, 5 Jun 2006 03:47:53 -0400 (EDT)
• References: <e5tqf3\$cvk\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Jeremy,

That is correct. If the eigenvalues are complex, they "appear" in 2X2
blocks along the diagonal of T in the decomposition A=Q.T.Q* . In your
case the eigenvalues are complex

mat={{-5.0, 7.0, 6.0, -3.0}, {2.0, -8.0, 3.0, 9.0}, \
{7.0, 1.0, 2.0, -7.0}, {3.0, 4.0, 4.0, 4.0}};

Eigenvalues[mat]

{-9.80269 + 0.336644 I, -9.80269 - 0.336644 I,
6.30269 + 4.81699 I, 6.30269 - 4.81699 I}

Now consider the Schur decomposition of mat

{Q, T} = SchurDecomposition[mat]

The T matrix is

T

{{6.30269,-6.10921,1.83689,5.59659},{3.79811,6.30269,

1.7072,6.28934},{0.,0.,-9.80269,6.03753},{0.,0.,-0.0187708,-9.80269}}

Thus in the first 2x2 block the diagonal values are the real part
(6.30269) and the off-diagonal terms can be used to construct the
imaginary part as follows:

Sqrt[T[[1,2]]*T[[2,1]]]

0. + 4.8169937117972985*I

Hope this helps,

Cheers,

Brian

{Q, T} = SchurDecomposition[]
Jeremy Watts wrote:
> why upon entering :-
>
> SchurDecomposition[{{-5.0, 7.0, 6.0, -3.0}, {2.0, -8.0, 3.0, 9.0}, {7.0,
> 1.0, 2.0, -7.0}, {3.0, 4.0, 4.0, 4.0}}]
> does mathematica return a 'T matrix' that is not upper triangular?
>
> some the eigenvalues are complex, so is it because these eigenvalues reside
> along the diagonal of 'T' in 2x2 blocks?

```

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