What are the limitations and capabilities of pseudo inverses?

  • Thread starter worryingchem
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In summary, the main difference between a pseudo inverse and a regular inverse is that a pseudo inverse can be used to solve for unknowns when there are more unknowns than equations. However, it is not unique and may have elements in the null space. The applications and drawbacks of using a pseudo inverse depend on the specific situation.
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worryingchem
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Hi, I was wondering what the major advantages and disadvantages of a pseudo inverse as compared to a regular inverse are.
In class, we're usually told to the number of equations have to match the number of unknowns to find a solution. I thought that the reason for that was so we can make a square matrix using those equations and calculate the unknown from taking the inverse. But after learning pseudo inverse, you can solve for unknowns when there are more unknowns than equations. So to restate my question, is there a limitation to the pseudo inverse, then? Can it do everything a regular inverse do?
 
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The limitation to the pseudo inverse is that it is not unique. You will have elements that are in the null space of the pseudo inverse operator.
Applications and drawbacks depend on your specific set up, but smartly applying pseudo inverses can be very useful.
 

1. What is the difference between inverse and pseudo inverse?

Inverse and pseudo inverse are both mathematical operations used to solve for parameters or variables in a system of equations. However, the main difference is that an inverse only exists for square matrices, while a pseudo inverse can also be computed for non-square matrices.

2. When should I use an inverse versus a pseudo inverse?

An inverse should be used when dealing with a square matrix and when a unique solution exists. A pseudo inverse should be used when dealing with a non-square matrix or when a unique solution does not exist. Pseudo inverse can also be used when dealing with ill-conditioned matrices.

3. How are an inverse and a pseudo inverse computed?

An inverse of a matrix is computed by using the determinant and the adjugate matrix. A pseudo inverse is computed by using the singular value decomposition (SVD) of the matrix.

4. Are an inverse and a pseudo inverse the same?

No, an inverse and a pseudo inverse are not the same. An inverse is the exact solution to a system of equations, while a pseudo inverse is an approximation of the solution.

5. Can either an inverse or a pseudo inverse be used to solve any type of system of equations?

No, an inverse can only be used for square matrices and when a unique solution exists. A pseudo inverse can be used for non-square matrices and when a unique solution does not exist, but it may not always provide an accurate solution.

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