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1 year ago in Algebra , Numerical Analysis By Fern G

How can one compute an updated matrix inverse when the last column is modified?

In my algorithm development for real-time parameter estimation, I have a known matrix inverse. When new data arrives, only the last column of the original matrix updates. Recomputing the full inverse is computationally expensive. I need a stable update formula or algorithm that leverages the existing inverse to find the new one more efficiently.

ConceptMatrix InverseFunctions Sherman-MorrisonFormula

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By Kanishka Answered 1 year ago

This is a classic application of the Sherman-Morrison-Woodbury formula for a rank-one update. If you have A?¹ and A changes only in its last column, you can express the change as the outer product of two vectors. The formula then gives the new inverse with O(n²) operations, far better than O(n³) for a full inversion. In practice, I would recommend careful implementation to check for numerical stability, as the denominator can become ill-conditioned. Always include a check for the validity of the update.

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