Prove that the limit of this matrix expression is 0

• I
• lriuui0x0
In summary, we are trying to prove that for a singular matrix A, the limit of (\chi_A(B) + \det(B)I)B^{-1} as t approaches 0 is equal to 0. This involves the product of a convergent to zero function and a divergent function, and we can use the Cayley-Hamilton theorem to show that \lim_{t\to 0} \chi_A(B) = \chi_A(A) = 0. We can also use the identity \operatorname{adj}(-A) = \sum_{n=0}^{N-1} a_{n+1}A^n to simplify the limit and show that it is either 2\
lriuui0x0
Given a singular matrix ##A##, let ##B = A - tI## for small positive ##t## such that ##B## is non-singular. Prove that:

$$\lim_{t\to 0} (\chi_A(B) + \det(B)I)B^{-1} = 0$$

where ##\chi_A## is the characteristic polynomial of ##A##. Note that ##\lim_{t\to 0} \chi_A(B) = \chi_A(A) = 0## by Cayley-Hamilton theorem.

This limit involves the product of a convergent to zero function and a divergent function. I'm not sure how to transform the limit in order to prove this.

In wikipedia characteristic polynomials
----
We consider an n×n matrix A. The characteristic polynomial of A, denoted by pA(t), is the polynomial defined by[5]
$${\displaystyle p_{A}(t)=\det \left(tI-A\right)}$$
where I denotes the n×n identity matrix.
----
Wiki says parameter of characteristic polynomial t a number though you say it B, a matrix. I am confused.

anuttarasammyak said:
In wikipedia characteristic polynomials
----
We consider an n×n matrix A. The characteristic polynomial of A, denoted by pA(t), is the polynomial defined by[5]
$${\displaystyle p_{A}(t)=\det \left(tI-A\right)}$$
where I denotes the n×n identity matrix.
----
Wiki says parameter of characteristic polynomial t a number though you say it B, a matrix. I am confused.
Yes, it's passing in the matrix into the polynomial. Please check Cayley-Hamilton theorem.

anuttarasammyak
I don't know how to tackle the problem, but here are some ideas:
• proof by induction
• replacing ##B^{-1}=\dfrac{1}{A-tI}## by its series
• using the Jordan normal form
• the limit has to be taken for every matrix entry, so maybe choosing a single one could simplify the problem

Let $$\chi_A(x) = \sum_{n=0}^N a_nx^n$$. Then $$(\chi_A(B) + \det(B)I)B^{-1} = (a_0 + \det B)B^{-1} + \sum_{n=0}^{N-1} a_{n+1}B^{n}.$$ Recall that $$B^{-1} = \frac{\operatorname{adj}(B)}{\det(B)}$$ where $\operatorname{adj}(B)$ is the adjugate matrix of $B$, and that $a_0$ vanishes as $A$ is singular. This shows that the limit is finite; there is more to do to show that it vanishes.

EDIT: We also have the identity (stated here) $$\operatorname{adj}(-A) = \sum_{n=0}^{N-1} a_{n+1}A^n$$ and thus $$\lim_{t \to 0} (\chi_A(B) + \det(B)I)B^{-1} = \operatorname{adj}(A) + \operatorname{adj}(-A) = (1 + (-1)^{N-1})\operatorname{adj}(A)$$ which is either $2\operatorname{adj}(A)$ or $0$ depending on whether $N$ is odd or even.

Last edited:
Thanks for the help!

1. How do I prove that the limit of a matrix expression is 0?

To prove that the limit of a matrix expression is 0, you can use the definition of a limit. This involves showing that for any small positive number, there exists a corresponding value of the matrix expression that is within that range. You can also use techniques such as the squeeze theorem or the use of epsilon-delta proofs.

2. Can I use L'Hopital's rule to prove the limit of a matrix expression is 0?

No, L'Hopital's rule only applies to limits involving functions, not matrices. You will need to use different techniques, such as the ones mentioned in the previous answer.

3. What are some common techniques for proving the limit of a matrix expression is 0?

Some common techniques for proving the limit of a matrix expression is 0 include using the definition of a limit, the squeeze theorem, and epsilon-delta proofs. You can also use properties of limits, such as the limit of a sum being equal to the sum of the limits.

4. Can I use numerical methods to prove the limit of a matrix expression is 0?

No, numerical methods are not suitable for proving the limit of a matrix expression is 0. These methods involve approximating the value of a limit, rather than providing a rigorous proof. It is important to use analytical techniques when proving mathematical statements.

5. Are there any special cases where the limit of a matrix expression may not be 0?

Yes, there are some special cases where the limit of a matrix expression may not be 0. For example, if the matrix expression involves a singularity (a value that makes the matrix non-invertible), then the limit may not exist. Additionally, if the limit of a matrix expression is undefined, then it cannot be proven to be 0.

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