Further Examples: PseudoInverse

Here is a matrix.

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The inverse does not exist.

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The pseudoinverse does exist.

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The pseudoinverse of a real matrix , usually denoted by , has the property that the sum of the squares of the entries of is minimized, where is an identity matrix of the appropriate size.

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For invertible matrices, the pseudoinverse is the same as the inverse.

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You can compute the pseudoinverse of a non-square matrix.

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This shows that pp satisfies the four conditions that uniquely characterize the pseudoinverse of an exact matrix.

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Using PseudoInverse to get minimal-length solutions of underdetermined systems

Here is a x matrix and a ­vector.

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The following system of equations is underdetermined.

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This finds a matrix whose columns consist of independent vectors in the nullspace of mat.

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This gives the minimal-length solution.

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This shows that the minimal­length solution is smaller than the original solution.

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