Information:
Equilibration (row/column scaling) is a preprocessing technique to improve matrix conditioning. For system , we multiply by diagonal matrices: , where .
How can equilibration reduce for a poorly scaled matrix (e.g., rows with vastly different magnitudes)?

Question

Information: 
 Equilibration  (row/column scaling) is a preprocessing technique to improve matrix conditioning. For system   , we multiply by diagonal matrices:   , where   . 
How can equilibration reduce    for a poorly scaled matrix (e.g., rows with vastly different magnitudes)?

Answer

Setting    achieves    exactly for any matrix via perfect conditioning

Answer

Equilibration cannot reduce    because condition number is invariant under diagonal scaling transformations

Answer

Equilibration increases    because added diagonal matrices introduce parameters that amplify errors

Answer

Normalizing row norms via    and column norms via    balances matrix entries, reducing the condition number effectively

Answer

Choosing    based on eigenvalues aligns the system with principal components to minimize

Answer

Setting    yields    with    trivially

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Information: Equilibration (row/column scaling) is a preprocessing technique ...

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