Question

Let \(U\) be a fixed \(n \times n\) matrix, and consider the operator \(T: \mathbf{M}_{n n} \rightarrow \mathbf{M}_{n n}\) given by \(T(A)=U A\). a. Show that \(\lambda\) is an eigenvalue of \(T\) if and only if it is an eigenvalue of \(U\) b. If \(\lambda\) is an eigenvalue of \(T\), show that \(E_{\lambda}(T)\) consists of all matrices whose columns lie in \(E_{\lambda}(U)\) : \(E_{\lambda}(T)\) \(=\left\\{\left[\begin{array}{llll}P_{1} & P_{2} & \cdots & P_{n}\end{array}\right] \mid P_{i}\right.\) in \(E_{\lambda}(U)\) for each \(\left.i\right\\}\) c. Show if \(\operatorname{dim}\left[E_{\lambda}(U)\right]=d\), then \(\operatorname{dim}\left[E_{\lambda}(T)\right]=n d\). [Hint: If \(B=\left\\{\mathbf{x}_{1}, \ldots, \mathbf{x}_{d}\right\\}\) is a basis of \(E_{\lambda}(U),\) consider the set of all matrices with one column from \(B\) and the other columns zero.]

Short answer

The eigenvalue \( \lambda \) of \( T \) is equivalent to an eigenvalue of \( U \). Eigenmatrices of \( T \) have columns in the eigenspace of \( U \), making \( \text{dim}(E_{\lambda}(T)) = n \times \text{dim}(E_{\lambda}(U)) \).

Step-by-step solution

Understand Eigenvalues and Eigenvectors

An eigenvalue \( \lambda \) of a matrix \( U \) is a scalar such that there exists a non-zero vector \( \mathbf{v} \) satisfying the equation \( U\mathbf{v} = \lambda\mathbf{v} \). In operator terms, \( T(A) = UA = \lambda A \) implies such matrices \( A \) are eigenvectors of \( T \).

Prove Part a - Eigenvalue Equivalence

To prove that \( \lambda \) is an eigenvalue of \( T \) if and only if it is an eigenvalue of \( U \):- If \( \lambda \) is an eigenvalue of \( T \), there is a non-zero matrix \( A \) such that \( T(A) = UA = \lambda A \). This relation implies that for any column vector \( \mathbf{v} \) of \( A \), \( U\mathbf{v} = \lambda \mathbf{v} \) holds, hence \( \lambda \) is also an eigenvalue of \( U \).- Conversely, if \( \lambda \) is an eigenvalue of \( U \), then there is a non-zero vector \( \mathbf{v} \) satisfying \( U\mathbf{v} = \lambda \mathbf{v} \). For any matrix \( A \) whose columns are composed of vectors such as \( \mathbf{v} \), \( UA = \lambda A \), proving \( \lambda \) is an eigenvalue of \( T \).

Prove Part b - Eigenmatrix Description

Given \( \lambda \) is an eigenvalue of \( T \), the eigenspace \( E_{\lambda}(T) \) consists of matrices whose columns are eigenvectors of \( U \) corresponding to eigenvalue \( \lambda \).- If \( A \in E_{\lambda}(T) \), each column \( \mathbf{v}_i \) of \( A \) satisfies \( U\mathbf{v}_i = \lambda \mathbf{v}_i \).- Hence, each column \( \mathbf{v}_i \) must lie in \( E_{\lambda}(U) \). Thus, \( E_{\lambda}(T) = \left\{ \left[ \begin{array}{llll} P_1 & P_2 & \cdots & P_n \end{array} \right] \mid P_i \in E_{\lambda}(U) \right\} \).

Build the Basis and Dimensionality

Assume \( B = \{ \mathbf{x}_1, \ldots, \mathbf{x}_d \} \) is a basis for \( E_{\lambda}(U) \), which is \( d \)-dimensional.- Construct matrices with one column from \( B \) and others zero. There are \( n \) columns and \( d \) choices for the non-zero column vector, totaling \( n \times d \) matrices.- These matrices form a basis for \( E_{\lambda}(T) \); thus, \( \text{dim}( E_{\lambda}(T) ) = n \times d \).

Key concepts

Linear Transformation

A linear transformation is a function between two vector spaces that preserves addition and scalar multiplication. It is like applying a formula that processes inputs to outputs, maintaining mathematical structure. Let's denote a linear transformation by the operator \( T \).- In this exercise, \( T \) is defined as \( T(A) = UA \), where \( U \) is a fixed \( n \times n \) matrix and \( A \) is any matrix in the space.- For a transformation to be linear, it must satisfy: 1. \( T(A + B) = T(A) + T(B) \) for any matrices \( A \) and \( B \). 2. \( T(cA) = cT(A) \) for any scalar \( c \) and matrix \( A \).Linear transformations frequently serve in mathematical modeling because they elegantly reflect real-world phenomena scaled or translated by constants.
They are foundational in various applications such as computer graphics, data analytics, and engineering optimization.

Matrix Operator

A matrix operator is an entity that processes matrices and outputs another matrix, implementing transformations by using matrix multiplication. Think of it as a machine that turns one matrix into another.- In our exercise, the matrix \( U \) acts as a matrix operator, impacting any matrix \( A \) by forming \( UA \).- This operation is fundamental because it influences how the modulated inputs (matrices in our function space) are transformed.Moreover, matrix operators are pivotal in describing systems of linear equations, which are solveable by these mathematical objects.- In programming and AI, matrix operators help perform large-scale calculations efficiently by utilizing the properties of linearity.
As matrix operators obey associative laws, they simplify complex algebraic operations making them accessible and practical in various calculations.

Eigenspace

The eigenspace of a linear transformation or matrix operator corresponding to an eigenvalue \( \lambda \) is the set of all eigenvectors associated with \( \lambda \), including the zero vector. It is a vector subspace rich with structure.- Eigenspaces help unravel the internal structure of matrices because they give standardized forms that simplify the matrix's operation or characterization.- In our specific case, \( E_{\lambda}(U) \) is the eigenspace of \( U \) for the eigenvalue \( \lambda \), composed of vectors satisfying \( U\mathbf{v} = \lambda \mathbf{v} \).- Similarly, \( E_{\lambda}(T) \) consists of matrices \( A \) whose columns fall within \( E_{\lambda}(U) \).This perspective allows mathematicians and engineers to simplify their problem resolutions using known eigenvectors and their combinations.
For practical applications, eigenspaces are deeply embedded in vibration analysis, stability studies, and system solvability examinations.

Matrix Dimension

The matrix dimension refers to the size of a matrix expressed as the number of rows and columns it contains. It's essential for understanding compatibility in operations like multiplication.- Understanding matrix dimension is critical because it directly influences aspects like computational complexity and data requirements.- In this context, if \( \text{dim}(E_{\lambda}(U)) = d \), \( d \) denotes the dimension of the eigenspace for eigenvalue \( \lambda \) in \( U \).- For the matrix operator \( T \), it expands to \( n \times d \), meaning the eigenspace dimension of \( T \) for \( \lambda \) now becomes \( n \times d \).Mathematicians care about matrix dimensions when optimizing algorithms and ensuring correct solution processes in computer science and engineering.- Accurate applications of matrix dimensions lead to more efficient resource use in simulations, automated learning systems, and digital image processing tasks.