If The Minimal Polynomial Of 𝑇 Is A Polynomial Multiple Of ( 𝑧 − 𝜆 ) 𝑚 (𝑧 − 𝜆)^𝑚 ( Z − 𝜆 ) M , Prove Dim N U L L ( 𝑇 − 𝜆 𝐼 ) 𝑚 Null(𝑇 − 𝜆𝐼)^𝑚 N U Ll ( T − 𝜆 I ) M ≥ 𝑚.

If the Minimal Polynomial of 𝑇 is a Polynomial Multiple of $(𝑧 − 𝜆)^𝑚$, Prove dim $null(𝑇 − 𝜆𝐼)^𝑚$ ≥ 𝑚

Introduction

In the realm of Linear Algebra, the concept of minimal polynomials plays a crucial role in understanding the properties of linear transformations. Given a linear transformation 𝑇 in a vector space 𝑉, the minimal polynomial of 𝑇 is the monic polynomial of smallest degree that annihilates 𝑇. In this article, we will explore the relationship between the minimal polynomial of 𝑇 and the dimension of the null space of $(𝑇 − 𝜆𝐼)^𝑚$, where 𝜆 is an eigenvalue of 𝑇 and 𝑚 is a positive integer.

Background

To begin with, let's recall some essential concepts. The null space of a linear transformation 𝑇, denoted by $null(𝑇)$, is the set of all vectors in the domain of 𝑇 that are mapped to the zero vector. The dimension of the null space, denoted by dim $null(𝑇)$, is the number of vectors in a basis for the null space.

The minimal polynomial of 𝑇 is a polynomial of the form $p(𝑧) = (𝑧 − 𝜆)^𝑚$, where 𝜆 is an eigenvalue of 𝑇 and 𝑚 is a positive integer. This polynomial is said to be a polynomial multiple of $(𝑧 − 𝜆)^𝑚$.

Proof

To prove that dim $null(𝑇 − 𝜆𝐼)^𝑚$ ≥ 𝑚, we will use the following steps:

Step 1: Establish the relationship between the minimal polynomial and the null space

Let $p(𝑧) = (𝑧 − 𝜆)^𝑚$ be the minimal polynomial of 𝑇. Then, we can write:

$$p(𝑇) = (𝑇 − 𝜆𝐼)^𝑚 = 0$$

This implies that the null space of $(𝑇 − 𝜆𝐼)^𝑚$ is non-trivial, i.e., $null((𝑇 − 𝜆𝐼)^𝑚) ≠ {0}$.

Step 2: Show that the null space of $(𝑇 − 𝜆𝐼)^𝑚$ has a basis of at least 𝑚 vectors

Let $\{v_1, v_2, ..., v_𝑚\}$ be a basis for the null space of $(𝑇 − 𝜆𝐼)^𝑚$. Then, we can write:

$$(𝑇 − 𝜆𝐼)^𝑚 v_i = 0$$

for each $i = 1, 2, ..., 𝑚$.

Step 3: Use the minimal polynomial to show that the vectors in the basis are linearly independent

Suppose that the vectors in the basis are linearly dependent. Then, there exist scalars $c_1, c_2, ..., c_𝑚$ such that:

$$c_1 v_1 + c_2 v_2 + ... + c_𝑚 v_𝑚 = 0$$

Applying the minimal polynomial to both sides, we get:

$$(𝑇 − 𝜆𝐼)^𝑚 (c_1 v_1 + c_2 v_2 + ... + c_𝑚 v_𝑚) = 0$$

Using the linearity of the transformation, we can rewrite this as:

$$c_1 (𝑇 − 𝜆𝐼)^𝑚 v_1 + c_2 (𝑇 − 𝜆𝐼)^𝑚 v_2 + ... + c_𝑚 (𝑇 − 𝜆𝐼)^𝑚 v_𝑚 = 0$$

Since $(𝑇 − 𝜆𝐼)^𝑚 v_i = 0$ for each $i = 1, 2, ..., 𝑚$, we have:

$$c_1 (0) + c_2 (0) + ... + c_𝑚 (0) = 0$$

This implies that $c_1 = c_2 = ... = c_𝑚 = 0$, which contradicts the assumption that the vectors in the basis are linearly dependent.

Step 4: Conclude that the dimension of the null space is at least 𝑚

Since the vectors in the basis are linearly independent, we have:

dim $null((𝑇 − 𝜆𝐼)^𝑚) ≥ 𝑚$

Conclusion

In this article, we have proven that if the minimal polynomial of 𝑇 is a polynomial multiple of $(𝑧 − 𝜆)^𝑚$, then the dimension of the null space of $(𝑇 − 𝜆𝐼)^𝑚$ is at least 𝑚. This result has important implications for understanding the properties of linear transformations and their minimal polynomials.

References

• Axler, S. (2015). Linear Algebra Done Right. Springer.
• Hoffman, K., & Kunze, R. (1971). Linear Algebra. Prentice Hall.

Note: The above article is a rewritten version of the original problem in Linear Algebra Done Right 4th edition, Exercise 8A Problem 4. The article is optimized for search engines and provides a clear and concise proof of the given statement.
Q&A: Minimal Polynomials and Null Spaces

Introduction

In our previous article, we explored the relationship between the minimal polynomial of a linear transformation 𝑇 and the dimension of the null space of $(𝑇 − 𝜆𝐼)^𝑚$. In this article, we will answer some frequently asked questions related to this topic.

Q1: What is the minimal polynomial of a linear transformation?

A1: The minimal polynomial of a linear transformation 𝑇 is the monic polynomial of smallest degree that annihilates 𝑇. In other words, it is the polynomial of smallest degree such that $p(𝑇) = 0$.

Q2: How is the minimal polynomial related to the null space of $(𝑇 − 𝜆𝐼)^𝑚$?

A2: If the minimal polynomial of 𝑇 is a polynomial multiple of $(𝑧 − 𝜆)^𝑚$, then the dimension of the null space of $(𝑇 − 𝜆𝐼)^𝑚$ is at least 𝑚.

Q3: What is the significance of the null space of $(𝑇 − 𝜆𝐼)^𝑚$?

A3: The null space of $(𝑇 − 𝜆𝐼)^𝑚$ is the set of all vectors in the domain of 𝑇 that are mapped to the zero vector by $(𝑇 − 𝜆𝐼)^𝑚$. This space is important because it provides information about the eigenvalues and eigenvectors of 𝑇.

Q4: How can we find the minimal polynomial of a linear transformation?

A4: To find the minimal polynomial of a linear transformation 𝑇, we can use the following steps:

1. Find the eigenvalues of 𝑇.
2. For each eigenvalue 𝜆, find the smallest positive integer 𝑚 such that $(𝑇 − 𝜆𝐼)^𝑚 = 0$.
3. The polynomial $(𝑧 − 𝜆)^𝑚$ is a factor of the minimal polynomial of 𝑇.

Q5: What is the relationship between the minimal polynomial and the characteristic polynomial of a linear transformation?

A5: The minimal polynomial of a linear transformation 𝑇 divides the characteristic polynomial of 𝑇. In other words, if $p(𝑧)$ is the minimal polynomial of 𝑇 and $c(𝑧)$ is the characteristic polynomial of 𝑇, then $p(𝑧)$ divides $c(𝑧)$.

Q6: How can we use the minimal polynomial to determine the eigenvalues of a linear transformation?

A6: To determine the eigenvalues of a linear transformation 𝑇, we can use the following steps:

1. Find the minimal polynomial of 𝑇.
2. Factor the minimal polynomial into linear factors.
3. The roots of the minimal polynomial are the eigenvalues of 𝑇.

Q7: What is the significance of the minimal polynomial in linear algebra?

A7: The minimal polynomial is an important concept in linear algebra because it provides information about the eigenvalues and eigenvectors of a linear transformation. It is also used to determine the dimension of the null space of a linear transformation.

Q8: Can the minimal polynomial be used to determine the rank of a linear transformation?

A8: Yes, the minimal polynomial can be used to determine the rank of a linear transformation. If the minimal polynomial of 𝑇 is $p(𝑧) = (𝑧 − 𝜆)^𝑚$, then the rank of 𝑇 is at most 𝑚.

Conclusion

In this article, we have answered some frequently asked questions related to the minimal polynomial of a linear transformation and its relationship to the null space of $(𝑇 − 𝜆𝐼)^𝑚$. We hope that this article has provided a clear and concise explanation of these concepts and has helped to clarify any confusion.

References

• Axler, S. (2015). Linear Algebra Done Right. Springer.
• Hoffman, K., & Kunze, R. (1971). Linear Algebra. Prentice Hall.