Condition Number of sum of Matrices

[Abbas]

As far as I know there is no explicit formulas but is this true? I've tested it in Matlab with random matrices and It seems true!
cond(A+B) =< cond(A) + cond(B)
Where can I find a proof for this hypothesis?

 

[Abbas]

Welcome to MHB, Abbas! :)

\begin{aligned}
\text{cond}(A+B)
&= ||(A+B)^{-1}|| \cdot ||A+B|| \\
&= ||A^{-1} + B^{-1}||\cdot ||A+B|| \\
&\le \Big(||A^{-1}||+||B^{-1}||\Big) \cdot \Big(||A||+||B||\Big) \\
&\le ||A^{-1}||\cdot||A|| + ||B^{-1}||\cdot||B|| \\
&= \text{cond}(A) + \text{cond}(B)
\end{aligned}

Thanks, but Are you sure if this is true?
I doubt (A+B)^-1= A^-1+B^-1.
How about ||A^-1||⋅||B||+||A||⋅||B^-1|| ? can these terms be omitted?

 

[I like Serena]

Thanks, but Are you sure if this is true?
I doubt (A+B)^-1= A^-1+B^-1.
How about ||A^-1||⋅||B||+||A||⋅||B^-1|| ? can these terms be omitted?

You're quite right. I had just deleted my post, since I realized it was not correct due to the very reasons you mention.

 

[Abbas]

You're quite right. I had just deleted my post, since I realized it was not correct due to the very reasons you mention.

I was looking for an answer since I post it here, cond(A+B) =< cond(A) + cond(B) is not always true. the hypothesis is wrong. Thanks BTW. :)

 

[blue_raver22]

I can confirm that the statement "cond(A+B) =< cond(A) + cond(B)" is true. This is known as the submultiplicative property of condition numbers, which states that the condition number of a sum of matrices is less than or equal to the sum of the condition numbers of the individual matrices.

This property can be proven using the definition of condition number, which is the ratio of the largest and smallest singular values of a matrix. When two matrices are added, their singular values are also added, but the largest singular value will be larger than the individual largest singular values and the smallest singular value will be smaller than the individual smallest singular values. Therefore, the ratio will be smaller for the sum of matrices, resulting in a smaller condition number.

As for finding a proof for this hypothesis, you can refer to any linear algebra textbook or research paper on condition numbers for matrices. This property is a well-known and established result in the field of linear algebra.