How can we proof this matrix norm equality?

In summary, a matrix norm is a function that assigns a non-negative value to a matrix, and a matrix norm equality refers to an equation stating that two different matrix norms are equal. Proving matrix norm equalities is important for understanding the relationship between different norms and for more efficient and accurate calculations. There are various methods for proving matrix norm equalities, and not all equalities are true for all types of matrices or conditions. Careful consideration of assumptions and limitations is necessary when using a matrix norm equality in calculations or proofs.
  • #1
JohnNL
1
0
||A-1|| = max ||x|| / ||Ax|| x[itex]\in[/itex]ℝn, x≠0 . x is a vector.
 
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  • #2
well try the reciprocal one, for the norm of A. (operator norm of course, assuming you know the definition. that's the place to start of course.)
 

1. What is a matrix norm?

A matrix norm is a function that assigns a non-negative value to a matrix, similar to how absolute value works for a scalar. It is used to measure the size of a matrix and can be thought of as a generalization of the concept of magnitude for vectors.

2. What is a matrix norm equality?

A matrix norm equality refers to an equation that states that two different matrix norms are equal. This means that the two norms measure the size of a matrix in the same way, despite potentially being calculated differently.

3. Why is it important to prove matrix norm equalities?

Proving matrix norm equalities is important because it helps us understand the relationship between different matrix norms and how they measure the size of matrices. It also allows us to make connections between seemingly different concepts and can lead to more efficient and accurate calculations.

4. How can we prove a matrix norm equality?

There are various methods for proving matrix norm equalities, including using properties of matrix norms, using mathematical induction, and using techniques from linear algebra such as matrix decompositions. It ultimately depends on the specific equality being proven and the available tools and techniques.

5. Are all matrix norm equalities true?

No, not all matrix norm equalities are true. Some may only hold for certain types of matrices or under specific conditions. It is important to carefully consider the assumptions and limitations of a given matrix norm equality before using it in calculations or proofs.

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