In this example, you will prove a key fact that we used in the proof of the convergence
of the iterative methods: ∥(I − B)−1∥ ≤ 1
1 − ∥B∥ if ∥B∥ < 1.
(a) It can be shown that I + B + B2 + B3 + ... is a well-defined quantity as long as
∥B∥ < 1.2 Prove that the inverse of I − B is I + B + B2 + B3 + .... (Just multiply
the two together and show that you get I.)
(b) Using the result of part (a), show that ∥(I − B)−1∥ ≤ 1 + ∥B∥ + ∥B∥2 + .... (Assume
that ∥I∥ = 1. This is not true for all norms, but it is true for all induced norms,
which are the norms we will usually use.)
(c) Finally, using the geometric series formula, ∞P
i=0
ri = 1
1−r for |r| < 1, conclude that
∥(I − B)−1∥ ≤ 1
1 − ∥B∥, if ∥B∥ < 1.
2You do not have to prove this.
3


To prove the key fact \(\| (I - B)^{-1} \| \leq \frac{1}{1 - \| B \|}\) for \(\| B \| < 1\), let's break the problem down into three parts as stated in your question.

### (a) Proving that \((I - B)^{-1} = I + B + B^2 + B^3 + \cdots\)

We are asked to show that the inverse of \(I - B\) is given by the infinite series:

\[
(I - B)^{-1} = I + B + B^2 + B^3 + \cdots
\]

This can be shown by multiplying \(I - B\) with the infinite sum \(I + B + B^2 + B^3 + \cdots\), and verifying that the result is the identity matrix \(I\).

Let:

\[
S = I + B + B^2 + B^3 + \cdots
\]

Now, calculate \( (I - B)S \):

\[
(I - B)S = (I - B)(I + B + B^2 + B^3 + \cdots)
\]

Distribute \(I - B\) over the terms:

\[
= I + B + B^2 + B^3 + \cdots - B(I + B + B^2 + B^3 + \cdots)
\]
\[
= I + B + B^2 + B^3 + \cdots - (B + B^2 + B^3 + B^4 + \cdots)
\]

Notice that all terms involving \(B\), \(B^2\), \(B^3\), etc., cancel out:

\[
= I
\]

Thus, we have shown that:

\[
(I - B)(I + B + B^2 + B^3 + \cdots) = I
\]

This means that the inverse of \(I - B\) is indeed \(I + B + B^2 + B^3 + \cdots\), so:

\[
(I - B)^{-1} = I + B + B^2 + B^3 + \cdots
\]

### (b) Showing that \(\| (I - B)^{-1} \| \leq 1 + \| B \| + \| B \|^2 + \cdots\)

Now that we know \( (I - B)^{-1} = I + B + B^2 + B^3 + \cdots \), we can take the norm of both sides. Assuming that \(\| \cdot \|\) is an induced norm, and that \(\| I \| = 1\) (which is true for induced norms):

\[
\| (I - B)^{-1} \| = \| I + B + B^2 + B^3 + \cdots \|
\]

Using the subadditivity of norms (i.e., \(\|A + B\| \leq \|A\| + \|B\|\)), we get:

\[
\| (I - B)^{-1} \| \leq \| I \| + \| B \| + \| B^2 \| + \| B^3 \| + \cdots
\]

Since \(\| I \| = 1\) and \(\| B^n \| \leq \| B \|^n\) for any matrix \(B\), this becomes:

\[
\| (I - B)^{-1} \| \leq 1 + \| B \| + \| B \|^2 + \| B \|^3 + \cdots
\]

### (c) Using the geometric series formula to conclude that \(\| (I - B)^{-1} \| \leq \frac{1}{1 - \| B \|}\)

The series \(1 + \| B \| + \| B \|^2 + \cdots\) is a geometric series with common ratio \(\| B \|\), and it converges as long as \(\| B \| < 1\).

Using the geometric series formula:

\[
\sum_{n=0}^{\infty} r^n = \frac{1}{1 - r} \quad \text{for} \quad |r| < 1
\]

we apply this formula with \(r = \| B \|\), which gives:

\[
1 + \| B \| + \| B \|^2 + \cdots = \frac{1}{1 - \| B \|}
\]

Therefore:

\[
\| (I - B)^{-1} \| \leq \frac{1}{1 - \| B \|}
\]

Thus, we have proven the desired inequality:

\[
\| (I - B)^{-1} \| \leq \frac{1}{1 - \| B \|} \quad \text{if} \quad \| B \| < 1
\]

### Conclusion

We have shown step by step that \(\| (I - B)^{-1} \| \leq \frac{1}{1 - \| B \|}\) for \(\| B \| < 1\), using properties of matrix norms and the geometric series formula.

---

Would you like any more details on any of the steps? Here are five related questions you might consider:

1. Can you show a specific example of a matrix \(B\) for which this result applies?
2. How does the result change if \(\| B \| \geq 1\)?
3. How does this proof relate to the convergence of iterative methods like Jacobi or Gauss-Seidel?
4. Can the geometric series expansion be applied to non-square matrices?
5. How does the choice of norm (e.g., spectral norm, Frobenius norm) affect the bounds in the proof?

**Tip**: When using norms in matrix analysis, the choice of norm (e.g., operator norm or Frobenius norm) can impact how inequalities involving matrix inverses are applied.

In this example, you will prove a key fact that we used in the proof of the convergence of the iterative methods: ∥(I − B)−1∥ ≤ 1 / (1 − ∥B∥) if ∥B∥ < 1. (a) It can be shown that I + B + B^2 + B^3 + ... is a well-defined quantity as long as ∥B∥ < 1. Prove that the inverse of I − B is I + B + B^2 + B^3 + .... (Just multiply the two together and show that you get I.) (b) Using the result of part (a), show that ∥(I − B)−1∥ ≤ 1 + ∥B∥ + ∥B∥^2 + .... (Assume that ∥I∥ = 1. This is not true for all norms, but it is true for all induced norms, which are the norms we will usually use.) (c) Finally, using the geometric series formula, ∑∞i=0 ri = 1 / (1−r) for |r| < 1, conclude that ∥(I − B)−1∥ ≤ 1 / (1 − ∥B∥), if ∥B∥ < 1.

This problem explores matrix inversion, norms, and the geometric series in proving the convergence of iterative methods. It includes step-by-step solutions and key theorems like the geometric series and matrix inversion, suitable for undergraduate students.

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