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melanie science forum beginner
Joined: 15 Jul 2005
Posts: 13

Posted: Mon Jun 26, 2006 7:33 pm Post subject:
Matrix problem; only brute force?



I have a question which I can not figure out
nice way of doing it except the brute force calculation.
I would be appreciated if someone could help me.
Let A be 2x2 matrix such that
[ 4 , 1 ]
[ 4 , 0 ]
find all alpha(real number) and
2x2 real synmetric matrix B such that
A* B* A_t = (alpha)* B
where A_t denotes transposition of A 

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Chip Eastham science forum Guru
Joined: 01 May 2005
Posts: 412

Posted: Mon Jun 26, 2006 7:58 pm Post subject:
Re: Matrix problem; only brute force?



melanie wrote:
Quote:  I have a question which I can not figure out
nice way of doing it except the brute force calculation.
I would be appreciated if someone could help me.
Let A be 2x2 matrix such that
[ 4 , 1 ]
[ 4 , 0 ]
find all alpha(real number) and
2x2 real synmetric matrix B such that
A* B* A_t = (alpha)* B
where A_t denotes transposition of A

Since A is a given matrix, it might be better
to rewrite the problem in this fashion:
Find all (real symmetric) B such that for
some (real scalar) alpha:
B^1 * A * B = (alpha) (A_t)^1
I'm not clear on your reason for restricting
B to be symmetric, but even for a general
matrix B to satisfy the condition above,
it must be that the spectrum (eigenvalues)
of A is the same as alpha times the set
of reciprocal eigenvalues of A (A and A_t,
its transpose, have the same spectrum).
In any case it narrows the possible values
of alpha considerably! Suppose A has
distinct eigenvalues r,s. Then either:
alpha = r^2 = s^2
or alpha = r*s.
regards, chip 

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Chip Eastham science forum Guru
Joined: 01 May 2005
Posts: 412

Posted: Mon Jun 26, 2006 8:04 pm Post subject:
Re: Matrix problem; only brute force?



Chip Eastham wrote:
Quote:  melanie wrote:
I have a question which I can not figure out
nice way of doing it except the brute force calculation.
I would be appreciated if someone could help me.
Let A be 2x2 matrix such that
[ 4 , 1 ]
[ 4 , 0 ]
find all alpha(real number) and
2x2 real synmetric matrix B such that
A* B* A_t = (alpha)* B
where A_t denotes transposition of A
Since A is a given matrix, it might be better
to rewrite the problem in this fashion:
Find all (real symmetric) B such that for
some (real scalar) alpha:
B^1 * A * B = (alpha) (A_t)^1
I'm not clear on your reason for restricting
B to be symmetric, but even for a general
matrix B to satisfy the condition above,
it must be that the spectrum (eigenvalues)
of A is the same as alpha times the set
of reciprocal eigenvalues of A (A and A_t,
its transpose, have the same spectrum).
In any case it narrows the possible values
of alpha considerably! Suppose A has
distinct eigenvalues r,s. Then either:
alpha = r^2 = s^2
or alpha = r*s.

Oops, I just noticed your particular A has
a double eigenvalue r = s = 2. So for the
specific matrix A =
[ 4 , 1 ]
[ 4 , 0 ]
you ask about, alpha must be 4. Sorry
for the oversight.
regards, chip 

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Ignacio Larrosa Cañestro1 science forum Guru Wannabe
Joined: 02 May 2005
Posts: 112

Posted: Mon Jun 26, 2006 8:07 pm Post subject:
Re: Matrix problem; only brute force?



En el
mensaje:33245839.1151350438870.JavaMail.jakarta@nitrogen.mathforum.org,
melanie <melanie@yahoo.com> escribió:
Quote:  I have a question which I can not figure out
nice way of doing it except the brute force calculation.
I would be appreciated if someone could help me.
Let A be 2x2 matrix such that
[ 4 , 1 ]
[ 4 , 0 ]
find all alpha(real number) and
2x2 real synmetric matrix B such that
A* B* A_t = (alpha)* B
where A_t denotes transposition of A

A = [4, 1; 4, 0], A_t = [4, 4; 1, 0], B = [x, y, z, t] ==>
A* B* A_t = [16x + 4y + 4z + t,  16x 4z;  16x  4y, 16x] = [kx, ky; kz,
kt]
(16k)x + 4y + 4z + t = 0
 16x  ky  4z = 0
 16x  4y  kz = 0
16x  kt = 0
But
C = [16k, 4, 4, 1; 16, k, 4, 0; 16, 4, k, 0; 16, 0, 0,  k] =
(k  4)^4
If k =/= 4, that system is homogeneus and determined ===> x = y = z 0 t = 0
is the only solution
If k = 4, the rank of C is 2, and you can set two values of {x, y, z, t} and
solve the others for it. By example,
x = t/4
y =  z  t

Best regards,
Ignacio Larrosa Cañestro
A Coruña (España)
ilarrosaQUITARMAYUSCULAS@mundor.com 

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Ignacio Larrosa Cañestro1 science forum Guru Wannabe
Joined: 02 May 2005
Posts: 112

Posted: Mon Jun 26, 2006 8:17 pm Post subject:
Re: Matrix problem; only brute force?



En el mensaje:4gat4oF1mfg3kU1@individual.net,
Ignacio Larrosa Cañestro <ilarrosaQUITARMAYUSCULAS@mundor.com> escribió:
Quote:  En el
mensaje:33245839.1151350438870.JavaMail.jakarta@nitrogen.mathforum.org,
melanie <melanie@yahoo.com> escribió:
I have a question which I can not figure out
nice way of doing it except the brute force calculation.
I would be appreciated if someone could help me.
Let A be 2x2 matrix such that
[ 4 , 1 ]
[ 4 , 0 ]
find all alpha(real number) and
2x2 real synmetric matrix B such that
A* B* A_t = (alpha)* B
where A_t denotes transposition of A
A = [4, 1; 4, 0], A_t = [4, 4; 1, 0], B = [x, y, z, t] ==
A* B* A_t = [16x + 4y + 4z + t,  16x 4z;  16x  4y, 16x] = [kx,
ky; kz, kt]
(16k)x + 4y + 4z + t = 0
 16x  ky  4z = 0
 16x  4y  kz = 0
16x  kt = 0
But
C = [16k, 4, 4, 1; 16, k, 4, 0; 16, 4, k, 0; 16, 0, 0, 
k] = (k  4)^4
If k =/= 4, that system is homogeneus and determined ===> x = y = z 0
t = 0 is the only solution
If k = 4, the rank of C is 2, and you can set two values of {x, y, z,
t} and solve the others for it. By example,
x = t/4
y =  z  t

Sorry

Best regards,
Ignacio Larrosa Cañestro
A Coruña (España)
ilarrosaQUITARMAYUSCULAS@mundor.com, I don't see the word 'symmetric'.
Then, simply do z = y. 

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Chip Eastham science forum Guru
Joined: 01 May 2005
Posts: 412

Posted: Mon Jun 26, 2006 11:35 pm Post subject:
Re: Matrix problem; only brute force?



Ignacio Larrosa Cañestro wrote:
Quote:  En el mensaje:4gat4oF1mfg3kU1@individual.net,
Ignacio Larrosa Cañestro <ilarrosaQUITARMAYUSCULAS@mundor.com> escribió:
En el
mensaje:33245839.1151350438870.JavaMail.jakarta@nitrogen.mathforum.org,
melanie <melanie@yahoo.com> escribió:
I have a question which I can not figure out
nice way of doing it except the brute force calculation.
I would be appreciated if someone could help me.
Let A be 2x2 matrix such that
[ 4 , 1 ]
[ 4 , 0 ]
find all alpha(real number) and
2x2 real synmetric matrix B such that
A* B* A_t = (alpha)* B
where A_t denotes transposition of A
A = [4, 1; 4, 0], A_t = [4, 4; 1, 0], B = [x, y, z, t] ==
A* B* A_t = [16x + 4y + 4z + t,  16x 4z;  16x  4y, 16x] = [kx,
ky; kz, kt]
(16k)x + 4y + 4z + t = 0
 16x  ky  4z = 0
 16x  4y  kz = 0
16x  kt = 0
But
C = [16k, 4, 4, 1; 16, k, 4, 0; 16, 4, k, 0; 16, 0, 0, 
k] = (k  4)^4
If k =/= 4, that system is homogeneus and determined ===> x = y = z 0
t = 0 is the only solution
If k = 4, the rank of C is 2, and you can set two values of {x, y, z,
t} and solve the others for it. By example,
x = t/4
y =  z  t
Sorry... I don't see the word 'symmetric'.
Then, simply do z = y.

Based on Igancio's analysis we can formulate a
recipe.
Let v be a right (column) eigenvector of A: Av = k*v
Let B = v v', where v' is the (row) transpose of v.
If v is a real vector, then B is real symmetric.
Now AB = kB, and by symmetry A B A_t = (k^2) B.
regards, chip 

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