Solving an equation through optimization..........

Y=Ax
 y is a mX1 matrix
 A is a mXn matrix
 x is a nX1 matrix
AA'=I

Here we apply optimization technique to solve x instead of taking inverse of A. Will you explain me why...............Please help me........

Answer 1

[Walter Roberson]

inverse is only strictly defined for square matrices.

You could use pinv() or the backslash operator ('\')

[Tinkul]

A is a square matrix.Even though A is a square matrix we apply optimization...Why.....

[Walter Roberson]

No, you said A is mxn which is not square unless m == n

[Tinkul]

Sorry,i applogy for that...m==n(ie, A is square matrix).Then why we use optimization...

[Walter Roberson]

http://blogs.mathworks.com/loren/2007/05/16/purpose-of-inv/

[Tinkul]

Sir, will you tell me why optimization algorithm is used to solve Y=Ax instead of taking A inverse.

[Walter Roberson]

http://blogs.mathworks.com/loren/2007/05/16/purpose-of-inv/#2

There are good reasons to not use inv however. The main one is

      numerical considerations.

Here is the second paragraph of the description for inv in the documentation:

In practice, it is seldom necessary to form the explicit inverse of a matrix. A frequent misuse of inv arises when solving the system of linear equations.

One way to solve this is with x = inv(A)*b. A better way, from both an execution time and numerical accuracy standpoint, is to use the matrix division operator x = A\b. This produces the solution using Gaussian elimination, without forming the inverse. See \ and / for further information.

[Walter Roberson]

So EXECUTION TIME and NUMERICAL ACCURACY.