Proving σmax(A-B) ≤ σmax(A) - σmin(B)

  • Thread starter hayu601
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In summary, σmax(A-B) represents the maximum singular value of the matrix resulting from the subtraction of matrix B from matrix A. It is calculated by finding the square root of the largest eigenvalue of the matrix (A-B)(A-B)ᵀ. σmax(A) - σmin(B) represents the difference between the maximum singular value of matrix A and the minimum singular value of matrix B. This inequality is important because it helps to understand the relationship between the singular values of two matrices and how they are affected by subtraction. It also has applications in fields such as control theory, signal processing, and data analysis.
  • #1
hayu601
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Is it true if I state:

σmax(A-B) <= σmax(A) - σmin(B) ?

I verify numerically that it is correct but how to prove it?
 
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  • #2
If [itex]a\in A[/itex] then [itex]a\le max(A)[/itex]
If [itex]b\in B[/itex] then [itex]b\ge min(a)[/itex] so [itex]-b\ge -min(B)[/itex]

Adding, [itex]a-b\le max(A)- min(B)[/itex] for all a in A, b in B/
 

1. What does σmax(A-B) represent?

σmax(A-B) represents the maximum singular value of the matrix resulting from the subtraction of matrix B from matrix A.

2. How is σmax(A-B) calculated?

σmax(A-B) is calculated by finding the square root of the largest eigenvalue of the matrix (A-B)(A-B)ᵀ.

3. What does σmax(A) - σmin(B) represent?

σmax(A) - σmin(B) represents the difference between the maximum singular value of matrix A and the minimum singular value of matrix B.

4. Why is it important to prove that σmax(A-B) ≤ σmax(A) - σmin(B)?

This inequality is important because it helps to understand the relationship between the singular values of two matrices and how they are affected by subtraction. It also allows us to make conclusions about the properties of the matrices involved.

5. What are some applications of this inequality in scientific research?

This inequality is commonly used in fields such as control theory, signal processing, and data analysis. It can also be applied in solving optimization problems and in studying the stability of linear systems.

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