Problem 6 - Algebra

Question

Let \(S\) be a set of \(n \times n\) matrices over a field \(k\). Show that there exists a column vector \(X \neq 0\) of dimension \(n\) in \(k\), such that \(M X=X\) for all \(M \in S\) if and only if there exists such a vector in some extension field \(k\) ' of \(k\).

Step-by-Step Solution

Answer

We have shown that there exists a column vector \(X \neq 0\) of dimension \(n\) in field \(k\), such that \(MX = X\) for all matrices \(M \in S\), if and only if there exists such a vector in some extension field \(k'\) of \(k\). This was proven by first assuming the existence of such a vector in field \(k\) and arguing its existence in an extension field \(k'\), and then assuming its existence in an extension field \(k'\) and showing it must also exist in field \(k\).

Step 1: Assume such a vector exists in field \(k\)

Let's assume that there exists a column vector \(X \neq 0\) of dimension \(n\) in field \(k\) such that \(MX = X\) for all matrices \(M \in S\).

Step 2: Show that such a vector exists in an extension field \(k'\) of \(k\)

As we assumed the existence of vector \(X \neq 0\) in field \(k\), it follows that \(X\) is also an element of any extension field \(k'\) of \(k\). Therefore, the equation \(MX = X\) still holds for all matrices \(M \in S\) in the extension field \(k'\) since every element of \(S\) is also an element of the set \(S\) defined over the extension field \(k'\).

Step 3: Assume such a vector exists in some extension field \(k'\) of \(k\)

Now let's assume that there exists a column vector \(X \neq 0\) of dimension \(n\) in some extension field \(k'\) of \(k\) such that the equation \(MX = X\) is satisfied for all matrices \(M \in S\) defined over the extension field \(k'\).

Step 4: Show that such a vector exists in field \(k\)

Consider the polynomial \(f(t)=\det(M - tI)\), where \(M \in S\). The coefficients of \(f(t)\) are in field \(k\), and it has at most \(n\) roots in any extension field of \(k\). Now let \(X\) be an eigenvector of \(M\) corresponding to an eigenvalue \(\lambda\) in some extension field \(k'\) of \(k\) (i.e., \(MX = \lambda X\)). Since \(\det(M - \lambda I) = 0\), \(\lambda\) must be a root of \(f(t)\), and thus belongs to \(k\). Since these properties hold for all matrices \(M \in S\), and all elements of \(S\) are also in \(k\), we can conclude that \(X\) exists in field \(k\).

Step 5: Conclusion

We have shown that there exists a column vector \(X \neq 0\) of dimension \(n\) in \(k\), such that \(M X=X\) for all \(M \in S\) if and only if there exists such a vector in some extension field \(k'\) of \(k\).

Key Concepts

Matrix Theory

Matrix theory is a fascinating branch of mathematics dedicated to the study of matrices, which are arrays of numbers arranged in rows and columns. In this area:

• A matrix can be multiplied by vectors to transform them, using operations like scaling and rotating.
• These operations look at how matrices interact and the properties they exhibit when combined with other matrices or vectors.

In our exercise, the focus is on finding a specific vector called an eigenvector that remains unchanged in direction after being multiplied by matrices from a set \(S\). This is a powerful concept because it simplifies many complex mathematical operations into more understandable forms.

Extension Fields

In mathematics, an extension field is a larger field that contains a smaller field as a subset.

• The idea is to expand the field to include solutions to equations that may not be present in the original field \(k\).
• This is crucial for certain matrix operations where the original field does not suffice to reveal all the properties of the matrices.

In the context of the exercise, the existence of an extension field \(k'\) assures us that even if a vector \(X\) cannot be found in the original field \(k\), it might exist in \(k'\). This means that if the vector properties hold in \(k'\), they must also hold in \(k\).

Linear Algebra

Linear algebra is the study of vectors, vector spaces, and linear transformations. It provides the framework for understanding how matrices operate and interact with vectors.

• The fundamental operations involve addition, scalar multiplication, and matrix multiplication.
• It enables the solving of linear systems, optimizations, and much more.

In our exercise, linear algebra facilitates finding whether a single vector can satisfy the condition \(MX = X\) for a set of matrices \(M\). This is a classic linear transformation problem where vectors remain invariant under such transformations.

Eigenvalues

Eigenvalues are scalars associated with a matrix that provide insightful properties about the matrix. These values indicate how much an eigenvector is stretched or shrunk during a transformation by a matrix.

• If \(MX = \lambda X\), \(\lambda\) is the eigenvalue corresponding to the eigenvector \(X\).
• Finding eigenvalues involves solving the characteristic polynomial \(\det(M - \lambda I) = 0\).

The step-by-step solution in the exercise shows that if a vector \(X\) can satisfy the condition in an extension field, the corresponding eigenvalue must reside in the original field \(k\). This ensures the concept's applicability across different fields, reinforcing its foundational role in matrix theory.