Solution of a matrix equation

technophobe · July 8, 2010, 18:25

Hi Guys,
I am currently doing some analysis that requires me to solve the following matrix equation:

ri = v * Ai * v'

where ri is a scalar, there are many of these (>n) and they are all known.
v is a row vector [v1 v2 ... vn], v' is it's transpose.
Ai is an n*n matrix corresponding to the ri values, these are all known.

The matrices Ai are symmetrical, real and all elements are positive.

I am trying to solve this equation for the vector v. Does anyone know of a simple method to solve this, or is anyone aware of online notes that may be able to help?

Many thanks,
B

idrama · July 10, 2010, 05:30

You have a number of equations of quatratic forms. You can use Newton Approch:

f_i(v)=ri - v.Ai.v'

F = [ f_1 , ... , f_n ]

N(v)=v-(DF^(-1)(v))*F(v)

where DF^(-1)(v) is the inverse of DF (gradient of F).
Start with some v*

July 8, 2010, 18:25	Solution of a matrix equation	#1
technophobe New Member Anon Join Date: Apr 2009 Posts: 13 Blog Entries: 1 Rep Power: 18	Hi Guys, I am currently doing some analysis that requires me to solve the following matrix equation: ri = v * Ai * v' where ri is a scalar, there are many of these (>n) and they are all known. v is a row vector [v1 v2 ... vn], v' is it's transpose. Ai is an n*n matrix corresponding to the ri values, these are all known. The matrices Ai are symmetrical, real and all elements are positive. I am trying to solve this equation for the vector v. Does anyone know of a simple method to solve this, or is anyone aware of online notes that may be able to help? Many thanks, B

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July 10, 2010, 05:30		#2
idrama Senior Member Claus Meister Join Date: Aug 2009 Location: Wiesbaden, Germany Posts: 241 Rep Power: 19	You have a number of equations of quatratic forms. You can use Newton Approch: f_i(v)=ri - v.Ai.v' F = [ f_1 , ... , f_n ] N(v)=v-(DF^(-1)(v))F(v) where DF^(-1)(v) is the inverse of DF (gradient of F). Start with some v