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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edited Jul 16 at 18:34

Rodrigo de Azevedo

12.5k41751

asked Jul 16 at 17:46

Roberto De La Fuente

417

• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  

add a comment | 

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

edited Jul 16 at 18:34

Rodrigo de Azevedo

12.5k41751

asked Jul 16 at 17:46

Roberto De La Fuente

417

• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  

add a comment | 

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

edited Jul 16 at 18:34

Rodrigo de Azevedo

12.5k41751

asked Jul 16 at 17:46

Roberto De La Fuente

417

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

edited Jul 16 at 18:34

Rodrigo de Azevedo

12.5k41751

asked Jul 16 at 17:46

Roberto De La Fuente

417

share|cite|improve this question

share|cite|improve this question

edited Jul 16 at 18:34

Rodrigo de Azevedo

12.5k41751

edited Jul 16 at 18:34

Rodrigo de Azevedo

12.5k41751

edited Jul 16 at 18:34

Rodrigo de Azevedo

12.5k41751

12.5k41751

asked Jul 16 at 17:46

Roberto De La Fuente

417

asked Jul 16 at 17:46

Roberto De La Fuente

417

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
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $F$ 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

add a comment | 

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

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

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

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

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

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

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

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

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

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

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

share|cite|improve this answer

answered Jul 16 at 18:05

Will Jagy

97.2k594196

• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  

add a comment | 

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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

answered Jul 16 at 18:05

Will Jagy

97.2k594196

• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  

add a comment | 

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

answered Jul 16 at 18:05

Will Jagy

97.2k594196

• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  

add a comment | 

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

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

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

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

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

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

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

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

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

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

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

share|cite|improve this answer

answered Jul 16 at 18:05

Will Jagy

97.2k594196

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

answered Jul 16 at 18:05

Will Jagy

97.2k594196

share|cite|improve this answer

share|cite|improve this answer

answered Jul 16 at 18:05

Will Jagy

97.2k594196

answered Jul 16 at 18:05

Will Jagy

97.2k594196

answered Jul 16 at 18:05

Will Jagy

97.2k594196

97.2k594196

• 
  
  
  

  
  

  
  
  
  
  
  

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

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  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

add a comment | 

 

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