Bilinear form and signature

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I have a bilinear form such that the associate matrix is



$$A = left(beginmatrix 0&0&k\ 0&k&0\ k&0&0endmatrixright)$$



For which $k$ the quadratic for has signature $(1,2)$?




matrices quadratic-forms bilinear-form




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  • $F$ is definited like $F: xz′+yy′+zx′$
    – Roberto De La Fuente
    Jul 16 at 17:47














up vote
0
down vote

favorite












I have a bilinear form such that the associate matrix is



$$A = left(beginmatrix 0&0&k\ 0&k&0\ k&0&0endmatrixright)$$



For which $k$ the quadratic for has signature $(1,2)$?




matrices quadratic-forms bilinear-form




share|cite|improve this question





















  • $F$ is definited like $F: xz′+yy′+zx′$
    – Roberto De La Fuente
    Jul 16 at 17:47












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I have a bilinear form such that the associate matrix is



$$A = left(beginmatrix 0&0&k\ 0&k&0\ k&0&0endmatrixright)$$



For which $k$ the quadratic for has signature $(1,2)$?




matrices quadratic-forms bilinear-form




share|cite|improve this question













I have a bilinear form such that the associate matrix is



$$A = left(beginmatrix 0&0&k\ 0&k&0\ k&0&0endmatrixright)$$



For which $k$ the quadratic for has signature $(1,2)$?





matrices quadratic-forms bilinear-form





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share|cite|improve this question




share|cite|improve this question








edited Jul 16 at 18:34









Rodrigo de Azevedo

12.5k41751




12.5k41751









asked Jul 16 at 17:46









Roberto De La Fuente

417




417











  • $F$ is definited like $F: xz′+yy′+zx′$
    – Roberto De La Fuente
    Jul 16 at 17:47
















  • $F$ is definited like $F: xz′+yy′+zx′$
    – Roberto De La Fuente
    Jul 16 at 17:47















$F$ is definited like $F: xz′+yy′+zx′$
– Roberto De La Fuente
Jul 16 at 17:47




$F$ is definited like $F: xz′+yy′+zx′$
– Roberto De La Fuente
Jul 16 at 17:47










1 Answer
1






active

oldest

votes

















up vote
0
down vote













This is all you need for Sylvester's Law of Inertia



$$ P^T H P = D $$
$$left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 1 \
- frac 1 2 & 0 & frac 1 2 \
endarray
right)
left(
beginarrayrrr
0 & 0 & k \
0 & k & 0 \
k & 0 & 0 \
endarray
right)
left(
beginarrayrrr
0 & 1 & - frac 1 2 \
1 & 0 & 0 \
0 & 1 & frac 1 2 \
endarray
right)
= left(
beginarrayrrr
k & 0 & 0 \
0 & 2 k& 0 \
0 & 0 & - frac k 2 \
endarray
right)
$$
$$ Q^T D Q = H $$
$$left(
beginarrayrrr
0 & frac 1 2 & - 1 \
1 & 0 & 0 \
0 & frac 1 2 & 1 \
endarray
right)
left(
beginarrayrrr
k & 0 & 0 \
0 & 2k & 0 \
0 & 0 & - frac k 2 \
endarray
right)
left(
beginarrayrrr
0 & 1 & 0 \
frac 1 2 & 0 & frac 1 2 \
- 1 & 0 & 1 \
endarray
right)
= left(
beginarrayrrr
0 & 0 & k \
0 & k & 0 \
k & 0 & 0 \
endarray
right)
$$



======================================================



Algorithm discussed at http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr
https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia

$$ H = left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
$$
$$ D_0 = H $$
$$ E_j^T D_j-1 E_j = D_j $$
$$ P_j-1 E_j = P_j $$
$$ E_j^-1 Q_j-1 = Q_j $$
$$ P_j Q_j = Q_j P_j = I $$
$$ P_j^T H P_j = D_j $$
$$ Q_j^T D_j Q_j = H $$



$$ H = left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
$$



==============================================



$$ E_1 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1 \
endarray
right)
$$
$$ P_1 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1 \
endarray
right)
, ; ; ; Q_1 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1 \
endarray
right)
, ; ; ; D_1 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 0 & 1 \
0 & 1 & 0 \
endarray
right)
$$



==============================================



$$ E_2 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 1 & 0 \
0 & 1 & 1 \
endarray
right)
$$
$$ P_2 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 1 & 1 \
endarray
right)
, ; ; ; Q_2 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
- 1 & 0 & 1 \
endarray
right)
, ; ; ; D_2 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 1 \
0 & 1 & 0 \
endarray
right)
$$



==============================================



$$ E_3 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 1 & - frac 1 2 \
0 & 0 & 1 \
endarray
right)
$$
$$ P_3 = left(
beginarrayrrr
0 & 1 & - frac 1 2 \
1 & 0 & 0 \
0 & 1 & frac 1 2 \
endarray
right)
, ; ; ; Q_3 = left(
beginarrayrrr
0 & 1 & 0 \
frac 1 2 & 0 & frac 1 2 \
- 1 & 0 & 1 \
endarray
right)
, ; ; ; D_3 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & - frac 1 2 \
endarray
right)
$$



==============================================



$$ P^T H P = D $$
$$left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 1 \
- frac 1 2 & 0 & frac 1 2 \
endarray
right)
left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
left(
beginarrayrrr
0 & 1 & - frac 1 2 \
1 & 0 & 0 \
0 & 1 & frac 1 2 \
endarray
right)
= left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & - frac 1 2 \
endarray
right)
$$
$$ Q^T D Q = H $$
$$left(
beginarrayrrr
0 & frac 1 2 & - 1 \
1 & 0 & 0 \
0 & frac 1 2 & 1 \
endarray
right)
left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & - frac 1 2 \
endarray
right)
left(
beginarrayrrr
0 & 1 & 0 \
frac 1 2 & 0 & frac 1 2 \
- 1 & 0 & 1 \
endarray
right)
= left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
$$






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  • i dont understand
    – Roberto De La Fuente
    Jul 16 at 18:16










Your Answer




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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
0
down vote













This is all you need for Sylvester's Law of Inertia



$$ P^T H P = D $$
$$left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 1 \
- frac 1 2 & 0 & frac 1 2 \
endarray
right)
left(
beginarrayrrr
0 & 0 & k \
0 & k & 0 \
k & 0 & 0 \
endarray
right)
left(
beginarrayrrr
0 & 1 & - frac 1 2 \
1 & 0 & 0 \
0 & 1 & frac 1 2 \
endarray
right)
= left(
beginarrayrrr
k & 0 & 0 \
0 & 2 k& 0 \
0 & 0 & - frac k 2 \
endarray
right)
$$
$$ Q^T D Q = H $$
$$left(
beginarrayrrr
0 & frac 1 2 & - 1 \
1 & 0 & 0 \
0 & frac 1 2 & 1 \
endarray
right)
left(
beginarrayrrr
k & 0 & 0 \
0 & 2k & 0 \
0 & 0 & - frac k 2 \
endarray
right)
left(
beginarrayrrr
0 & 1 & 0 \
frac 1 2 & 0 & frac 1 2 \
- 1 & 0 & 1 \
endarray
right)
= left(
beginarrayrrr
0 & 0 & k \
0 & k & 0 \
k & 0 & 0 \
endarray
right)
$$



======================================================



Algorithm discussed at http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr
https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia

$$ H = left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
$$
$$ D_0 = H $$
$$ E_j^T D_j-1 E_j = D_j $$
$$ P_j-1 E_j = P_j $$
$$ E_j^-1 Q_j-1 = Q_j $$
$$ P_j Q_j = Q_j P_j = I $$
$$ P_j^T H P_j = D_j $$
$$ Q_j^T D_j Q_j = H $$



$$ H = left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
$$



==============================================



$$ E_1 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1 \
endarray
right)
$$
$$ P_1 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1 \
endarray
right)
, ; ; ; Q_1 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1 \
endarray
right)
, ; ; ; D_1 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 0 & 1 \
0 & 1 & 0 \
endarray
right)
$$



==============================================



$$ E_2 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 1 & 0 \
0 & 1 & 1 \
endarray
right)
$$
$$ P_2 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 1 & 1 \
endarray
right)
, ; ; ; Q_2 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
- 1 & 0 & 1 \
endarray
right)
, ; ; ; D_2 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 1 \
0 & 1 & 0 \
endarray
right)
$$



==============================================



$$ E_3 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 1 & - frac 1 2 \
0 & 0 & 1 \
endarray
right)
$$
$$ P_3 = left(
beginarrayrrr
0 & 1 & - frac 1 2 \
1 & 0 & 0 \
0 & 1 & frac 1 2 \
endarray
right)
, ; ; ; Q_3 = left(
beginarrayrrr
0 & 1 & 0 \
frac 1 2 & 0 & frac 1 2 \
- 1 & 0 & 1 \
endarray
right)
, ; ; ; D_3 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & - frac 1 2 \
endarray
right)
$$



==============================================



$$ P^T H P = D $$
$$left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 1 \
- frac 1 2 & 0 & frac 1 2 \
endarray
right)
left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
left(
beginarrayrrr
0 & 1 & - frac 1 2 \
1 & 0 & 0 \
0 & 1 & frac 1 2 \
endarray
right)
= left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & - frac 1 2 \
endarray
right)
$$
$$ Q^T D Q = H $$
$$left(
beginarrayrrr
0 & frac 1 2 & - 1 \
1 & 0 & 0 \
0 & frac 1 2 & 1 \
endarray
right)
left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & - frac 1 2 \
endarray
right)
left(
beginarrayrrr
0 & 1 & 0 \
frac 1 2 & 0 & frac 1 2 \
- 1 & 0 & 1 \
endarray
right)
= left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
$$






share|cite|improve this answer





















  • i dont understand
    – Roberto De La Fuente
    Jul 16 at 18:16














up vote
0
down vote













This is all you need for Sylvester's Law of Inertia



$$ P^T H P = D $$
$$left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 1 \
- frac 1 2 & 0 & frac 1 2 \
endarray
right)
left(
beginarrayrrr
0 & 0 & k \
0 & k & 0 \
k & 0 & 0 \
endarray
right)
left(
beginarrayrrr
0 & 1 & - frac 1 2 \
1 & 0 & 0 \
0 & 1 & frac 1 2 \
endarray
right)
= left(
beginarrayrrr
k & 0 & 0 \
0 & 2 k& 0 \
0 & 0 & - frac k 2 \
endarray
right)
$$
$$ Q^T D Q = H $$
$$left(
beginarrayrrr
0 & frac 1 2 & - 1 \
1 & 0 & 0 \
0 & frac 1 2 & 1 \
endarray
right)
left(
beginarrayrrr
k & 0 & 0 \
0 & 2k & 0 \
0 & 0 & - frac k 2 \
endarray
right)
left(
beginarrayrrr
0 & 1 & 0 \
frac 1 2 & 0 & frac 1 2 \
- 1 & 0 & 1 \
endarray
right)
= left(
beginarrayrrr
0 & 0 & k \
0 & k & 0 \
k & 0 & 0 \
endarray
right)
$$



======================================================



Algorithm discussed at http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr
https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia

$$ H = left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
$$
$$ D_0 = H $$
$$ E_j^T D_j-1 E_j = D_j $$
$$ P_j-1 E_j = P_j $$
$$ E_j^-1 Q_j-1 = Q_j $$
$$ P_j Q_j = Q_j P_j = I $$
$$ P_j^T H P_j = D_j $$
$$ Q_j^T D_j Q_j = H $$



$$ H = left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
$$



==============================================



$$ E_1 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1 \
endarray
right)
$$
$$ P_1 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1 \
endarray
right)
, ; ; ; Q_1 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1 \
endarray
right)
, ; ; ; D_1 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 0 & 1 \
0 & 1 & 0 \
endarray
right)
$$



==============================================



$$ E_2 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 1 & 0 \
0 & 1 & 1 \
endarray
right)
$$
$$ P_2 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 1 & 1 \
endarray
right)
, ; ; ; Q_2 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
- 1 & 0 & 1 \
endarray
right)
, ; ; ; D_2 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 1 \
0 & 1 & 0 \
endarray
right)
$$



==============================================



$$ E_3 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 1 & - frac 1 2 \
0 & 0 & 1 \
endarray
right)
$$
$$ P_3 = left(
beginarrayrrr
0 & 1 & - frac 1 2 \
1 & 0 & 0 \
0 & 1 & frac 1 2 \
endarray
right)
, ; ; ; Q_3 = left(
beginarrayrrr
0 & 1 & 0 \
frac 1 2 & 0 & frac 1 2 \
- 1 & 0 & 1 \
endarray
right)
, ; ; ; D_3 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & - frac 1 2 \
endarray
right)
$$



==============================================



$$ P^T H P = D $$
$$left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 1 \
- frac 1 2 & 0 & frac 1 2 \
endarray
right)
left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
left(
beginarrayrrr
0 & 1 & - frac 1 2 \
1 & 0 & 0 \
0 & 1 & frac 1 2 \
endarray
right)
= left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & - frac 1 2 \
endarray
right)
$$
$$ Q^T D Q = H $$
$$left(
beginarrayrrr
0 & frac 1 2 & - 1 \
1 & 0 & 0 \
0 & frac 1 2 & 1 \
endarray
right)
left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & - frac 1 2 \
endarray
right)
left(
beginarrayrrr
0 & 1 & 0 \
frac 1 2 & 0 & frac 1 2 \
- 1 & 0 & 1 \
endarray
right)
= left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
$$






share|cite|improve this answer





















  • i dont understand
    – Roberto De La Fuente
    Jul 16 at 18:16












up vote
0
down vote










up vote
0
down vote









This is all you need for Sylvester's Law of Inertia



$$ P^T H P = D $$
$$left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 1 \
- frac 1 2 & 0 & frac 1 2 \
endarray
right)
left(
beginarrayrrr
0 & 0 & k \
0 & k & 0 \
k & 0 & 0 \
endarray
right)
left(
beginarrayrrr
0 & 1 & - frac 1 2 \
1 & 0 & 0 \
0 & 1 & frac 1 2 \
endarray
right)
= left(
beginarrayrrr
k & 0 & 0 \
0 & 2 k& 0 \
0 & 0 & - frac k 2 \
endarray
right)
$$
$$ Q^T D Q = H $$
$$left(
beginarrayrrr
0 & frac 1 2 & - 1 \
1 & 0 & 0 \
0 & frac 1 2 & 1 \
endarray
right)
left(
beginarrayrrr
k & 0 & 0 \
0 & 2k & 0 \
0 & 0 & - frac k 2 \
endarray
right)
left(
beginarrayrrr
0 & 1 & 0 \
frac 1 2 & 0 & frac 1 2 \
- 1 & 0 & 1 \
endarray
right)
= left(
beginarrayrrr
0 & 0 & k \
0 & k & 0 \
k & 0 & 0 \
endarray
right)
$$



======================================================



Algorithm discussed at http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr
https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia

$$ H = left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
$$
$$ D_0 = H $$
$$ E_j^T D_j-1 E_j = D_j $$
$$ P_j-1 E_j = P_j $$
$$ E_j^-1 Q_j-1 = Q_j $$
$$ P_j Q_j = Q_j P_j = I $$
$$ P_j^T H P_j = D_j $$
$$ Q_j^T D_j Q_j = H $$



$$ H = left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
$$



==============================================



$$ E_1 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1 \
endarray
right)
$$
$$ P_1 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1 \
endarray
right)
, ; ; ; Q_1 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1 \
endarray
right)
, ; ; ; D_1 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 0 & 1 \
0 & 1 & 0 \
endarray
right)
$$



==============================================



$$ E_2 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 1 & 0 \
0 & 1 & 1 \
endarray
right)
$$
$$ P_2 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 1 & 1 \
endarray
right)
, ; ; ; Q_2 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
- 1 & 0 & 1 \
endarray
right)
, ; ; ; D_2 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 1 \
0 & 1 & 0 \
endarray
right)
$$



==============================================



$$ E_3 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 1 & - frac 1 2 \
0 & 0 & 1 \
endarray
right)
$$
$$ P_3 = left(
beginarrayrrr
0 & 1 & - frac 1 2 \
1 & 0 & 0 \
0 & 1 & frac 1 2 \
endarray
right)
, ; ; ; Q_3 = left(
beginarrayrrr
0 & 1 & 0 \
frac 1 2 & 0 & frac 1 2 \
- 1 & 0 & 1 \
endarray
right)
, ; ; ; D_3 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & - frac 1 2 \
endarray
right)
$$



==============================================



$$ P^T H P = D $$
$$left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 1 \
- frac 1 2 & 0 & frac 1 2 \
endarray
right)
left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
left(
beginarrayrrr
0 & 1 & - frac 1 2 \
1 & 0 & 0 \
0 & 1 & frac 1 2 \
endarray
right)
= left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & - frac 1 2 \
endarray
right)
$$
$$ Q^T D Q = H $$
$$left(
beginarrayrrr
0 & frac 1 2 & - 1 \
1 & 0 & 0 \
0 & frac 1 2 & 1 \
endarray
right)
left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & - frac 1 2 \
endarray
right)
left(
beginarrayrrr
0 & 1 & 0 \
frac 1 2 & 0 & frac 1 2 \
- 1 & 0 & 1 \
endarray
right)
= left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
$$






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This is all you need for Sylvester's Law of Inertia



$$ P^T H P = D $$
$$left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 1 \
- frac 1 2 & 0 & frac 1 2 \
endarray
right)
left(
beginarrayrrr
0 & 0 & k \
0 & k & 0 \
k & 0 & 0 \
endarray
right)
left(
beginarrayrrr
0 & 1 & - frac 1 2 \
1 & 0 & 0 \
0 & 1 & frac 1 2 \
endarray
right)
= left(
beginarrayrrr
k & 0 & 0 \
0 & 2 k& 0 \
0 & 0 & - frac k 2 \
endarray
right)
$$
$$ Q^T D Q = H $$
$$left(
beginarrayrrr
0 & frac 1 2 & - 1 \
1 & 0 & 0 \
0 & frac 1 2 & 1 \
endarray
right)
left(
beginarrayrrr
k & 0 & 0 \
0 & 2k & 0 \
0 & 0 & - frac k 2 \
endarray
right)
left(
beginarrayrrr
0 & 1 & 0 \
frac 1 2 & 0 & frac 1 2 \
- 1 & 0 & 1 \
endarray
right)
= left(
beginarrayrrr
0 & 0 & k \
0 & k & 0 \
k & 0 & 0 \
endarray
right)
$$



======================================================



Algorithm discussed at http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr
https://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia

$$ H = left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
$$
$$ D_0 = H $$
$$ E_j^T D_j-1 E_j = D_j $$
$$ P_j-1 E_j = P_j $$
$$ E_j^-1 Q_j-1 = Q_j $$
$$ P_j Q_j = Q_j P_j = I $$
$$ P_j^T H P_j = D_j $$
$$ Q_j^T D_j Q_j = H $$



$$ H = left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
$$



==============================================



$$ E_1 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1 \
endarray
right)
$$
$$ P_1 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1 \
endarray
right)
, ; ; ; Q_1 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 0 & 1 \
endarray
right)
, ; ; ; D_1 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 0 & 1 \
0 & 1 & 0 \
endarray
right)
$$



==============================================



$$ E_2 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 1 & 0 \
0 & 1 & 1 \
endarray
right)
$$
$$ P_2 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
0 & 1 & 1 \
endarray
right)
, ; ; ; Q_2 = left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 0 \
- 1 & 0 & 1 \
endarray
right)
, ; ; ; D_2 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 1 \
0 & 1 & 0 \
endarray
right)
$$



==============================================



$$ E_3 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 1 & - frac 1 2 \
0 & 0 & 1 \
endarray
right)
$$
$$ P_3 = left(
beginarrayrrr
0 & 1 & - frac 1 2 \
1 & 0 & 0 \
0 & 1 & frac 1 2 \
endarray
right)
, ; ; ; Q_3 = left(
beginarrayrrr
0 & 1 & 0 \
frac 1 2 & 0 & frac 1 2 \
- 1 & 0 & 1 \
endarray
right)
, ; ; ; D_3 = left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & - frac 1 2 \
endarray
right)
$$



==============================================



$$ P^T H P = D $$
$$left(
beginarrayrrr
0 & 1 & 0 \
1 & 0 & 1 \
- frac 1 2 & 0 & frac 1 2 \
endarray
right)
left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
left(
beginarrayrrr
0 & 1 & - frac 1 2 \
1 & 0 & 0 \
0 & 1 & frac 1 2 \
endarray
right)
= left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & - frac 1 2 \
endarray
right)
$$
$$ Q^T D Q = H $$
$$left(
beginarrayrrr
0 & frac 1 2 & - 1 \
1 & 0 & 0 \
0 & frac 1 2 & 1 \
endarray
right)
left(
beginarrayrrr
1 & 0 & 0 \
0 & 2 & 0 \
0 & 0 & - frac 1 2 \
endarray
right)
left(
beginarrayrrr
0 & 1 & 0 \
frac 1 2 & 0 & frac 1 2 \
- 1 & 0 & 1 \
endarray
right)
= left(
beginarrayrrr
0 & 0 & 1 \
0 & 1 & 0 \
1 & 0 & 0 \
endarray
right)
$$







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share|cite|improve this answer



share|cite|improve this answer











answered Jul 16 at 18:05









Will Jagy

97.2k594196




97.2k594196











  • i dont understand
    – Roberto De La Fuente
    Jul 16 at 18:16
















  • i dont understand
    – Roberto De La Fuente
    Jul 16 at 18:16















i dont understand
– Roberto De La Fuente
Jul 16 at 18:16




i dont understand
– Roberto De La Fuente
Jul 16 at 18:16












 

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