Question

Let \(T: V \rightarrow V\) be a linear operator where \(\operatorname{dim} V=n\). Show that \(V\) has a basis of eigenvectors if and only if \(V\) has a basis \(B\) such that \(M_{B}(T)\) is diagonal.

Short answer

A basis of eigenvectors exists if, and only if, the matrix \(M_B(T)\) is diagonal.

Step-by-step solution

Understand the Problem Statement

We need to show the equivalence between two statements about a linear operator \(T\) on vector space \(V\): 1) that \(V\) has a basis of eigenvectors, and 2) there exists a basis \(B\) such that the matrix of \(T\) relative to \(B\) is diagonal.

Recall Definitions

A basis of eigenvectors for a vector space \(V\) is a set of linearly independent vectors such that each vector is an eigenvector of \(T\). For a matrix to be diagonal, it means it has non-zero entries only on its diagonal.

Show 'Eigenvectors Implies Diagonal Matrix'

Assume \(V\) has a basis of eigenvectors \(\{v_1, v_2, \ldots, v_n\}\) corresponding to eigenvalues \(\{\lambda_1, \lambda_2, \ldots, \lambda_n\}\). Construct the matrix \(M_B(T)\) using this basis. By definition, \(T(v_i) = \lambda_i v_i\). Thus, the matrix is diagonal with \(\lambda_i\) as the diagonal entries.

Show 'Diagonal Matrix Implies Eigenvectors'

Assume \(M_B(T)\) is diagonal with diagonal entries \(\lambda_1, \lambda_2, \ldots, \lambda_n\). Each diagonal entry corresponds to an eigenvalue. Since \(M_B(T)(e_i) = \lambda_i e_i\), where \(e_i\) are the basis vectors, \(e_i\) are eigenvectors of \(T\). Thus, \(\{e_1, e_2, \ldots, e_n\}\) forms a basis of eigenvectors.

Conclude Equivalence

Both directions have been shown: if \(V\) has a basis of eigenvectors, then \(M_B(T)\) is diagonal, and if \(M_B(T)\) is diagonal, then \(V\) has a basis of eigenvectors. This proves the equivalence.

Key concepts

Eigenvectors

Eigenvectors are special vectors associated with a linear transformation. When a vector space has a linear operator, the eigenvectors are the vectors that only get scaled by the transformation. This means if we apply the operator to an eigenvector, we simply multiply it by a unique number called the eigenvalue. Consider the effect of stretching or compressing an object—yet keeping its orientation intact. The key features of eigenvectors include:

• They do not change direction under the transformation.
• They are crucial in forming the basis of a vector space.

Because of these properties, if you can find a set of linearly independent eigenvectors for every vector in space, you can dramatically simplify complex transformations.

Diagonal Matrix

A diagonal matrix is a matrix that has non-zero elements along its main diagonal, with all other elements being zero. This form is incredibly nice to work with due to its highly simplified structure. For example, consider a matrix:\[\begin{bmatrix}\lambda_1 & 0 & \cdots & 0 \0 & \lambda_2 & \cdots & 0 \\vdots & \vdots & \ddots & \vdots \0 & 0 & \cdots & \lambda_n\end{bmatrix}\]It’s easy to compute powers of diagonal matrices because it involves simply raising each diagonal entry to the power. Diagonalizing a matrix makes complex calculations more manageable:

• It simplifies power calculations of the matrix.
• It highlights eigenvalues directly on the diagonal.

In linear algebra, transforming a matrix to a diagonal form is akin to clearing the clutter—it allows for straightforward interpretation of a transformation’s effect.

Vector Space

In linear algebra, a vector space is a fundamental concept used to describe a collection of vectors that can be scaled and added together. Imagine different roads branching out infinitely along directions—this represents vectors within the space. A vector space is characterized by:

• Closure under addition and scalar multiplication.
• A zero vector that does not change any vector when added to it.
• Associated properties such as commutativity and associativity.

Understanding the structure of a vector space helps in grasping more complex concepts like transformations and basis, being the canvas upon which these mathematical ideas are painted.

Basis

A basis is like the foundation of a vector space—it's a specific set of vectors from which all other vectors in the space can be constructed through linear combinations. Think of basis vectors as the building blocks of the vector space.

• The vectors in a basis are linearly independent and span the whole vector space.
• A vector space can have many different bases, but all have the same number of vectors.

In the context of a linear operator, having a basis of eigenvectors implies that the operator is diagonizable. This is pivotal for simplifying matrix operations and gaining deeper insights into the vector space's geometry

Eigenvalues

Eigenvalues are the constants associated with eigenvectors, showing how much the eigenvector is stretched or compressed during a linear transformation. If you have an eigenvector \(v\) such that applying the transformation yields \(Tv = \lambda v\), then \(\lambda\) is the eigenvalue. Key characteristics of eigenvalues include:

• They provide insight into the "magnitude" of the transformation effect.
• They directly appear as entries in a diagonal matrix when a transformation matrix is diagonalizable.

Similar to eigenvectors, finding all eigenvalues of a transformation offers a comprehensive understanding of how the transformation behaves over a span of a vector space. This understanding is crucial for applications ranging from stability analysis to quantum mechanics.