Question

We often write vectors in \(\mathbb{R}^{n}\) as rows. Let \(P\) denote an invertible \(n \times n\) matrix. If \(\lambda\) is a number, show that $$ E_{\lambda}\left(P A P^{-1}\right)=\left\\{P \mathbf{x} \mid \mathbf{x} \text { is in } E_{\lambda}(A)\right\\} $$ for each \(n \times n\) matrix \(A\).

Short answer

The eigenspace transformation is given by \( E_{\lambda}(P A P^{-1}) = \{ P \mathbf{x} \mid \mathbf{x} \in E_{\lambda}(A) \} \).

Step-by-step solution

Understand Eigenvectors and Eigenvalues

Recall that an eigenvector \( \mathbf{x} \) of a matrix \( A \) corresponding to an eigenvalue \( \lambda \) satisfies the equation \( A \mathbf{x} = \lambda \mathbf{x} \). We denote the eigenspace of \( A \) corresponding to \( \lambda \) as \( E_{\lambda}(A) \). This is the set of all eigenvectors associated with \( \lambda \) along with the zero vector.

Set Up the Transformation with P

Consider the new matrix \( B = P A P^{-1} \). We want to find the eigenspace \( E_{\lambda}(B) \) corresponding to \( \lambda \). Assume \( \mathbf{y} \) is an eigenvector of \( B \) corresponding to \( \lambda \). Then, \( B \mathbf{y} = \lambda \mathbf{y} \).

Substitute B with P A P^{-1}

Using the definition of \( B \), substitute to get \( (P A P^{-1}) \mathbf{y} = \lambda \mathbf{y} \). Multiply both sides by \( P^{-1} \), yielding \( A (P^{-1} \mathbf{y}) = \lambda (P^{-1} \mathbf{y}) \). This implies that \( P^{-1} \mathbf{y} \) is an eigenvector of \( A \).

Express y in terms of x

Let \( \mathbf{x} = P^{-1} \mathbf{y} \). Since \( A \mathbf{x} = \lambda \mathbf{x} \), it follows that \( \mathbf{x} \) is in \( E_{\lambda}(A) \). Consequently, \( \mathbf{y} = P \mathbf{x} \). Substitute \( \mathbf{x} \) back: \( \mathbf{y} = P \mathbf{x} \) where \( \mathbf{x} \) belongs to \( E_{\lambda}(A) \).

Conclude the Eigenspace Transformation

From Step 4, we conclude that each eigenvector \( \mathbf{y} \) of \( B = P A P^{-1} \) is of the form \( P \mathbf{x} \), where \( \mathbf{x} \) is an eigenvector of \( A \) for the same eigenvalue \( \lambda \). Hence, \( E_{\lambda}(P A P^{-1}) = \{ P \mathbf{x} \mid \mathbf{x} \in E_{\lambda}(A) \} \), as required.

Key concepts

Matrix Transformations

Matrix transformations are operations that change vectors in a systematic way using matrices. It's like pressing a button on a machine that rotates, scales, or flips objects in space. These transformations can be represented as multiplications of matrices with vectors. - **Types of Transformations**: - **Scaling**: Changes the size of objects. - **Rotation**: Spins objects around a point. - **Reflection**: Flips objects over a mirror line. - **Translation**: Moves objects in space.When dealing with transformations, an important concept is that of similarity transformations, which involve multiplying a matrix by other matrices, such as \( P A P^{-1} \). This specific transformation keeps the core properties of the matrix \( A \) intact, like its eigenvalues, while presenting the matrix in a possibly simpler or more useful form. These transformations are crucial in simplifying complex matrices and in changing the basis of vector spaces, allowing us to view data or problems from different perspectives.

Invertible Matrices

An invertible matrix, or a non-singular matrix, is a square matrix that has a multiplicative inverse. - **Key Characteristics**: - Must be square (same number of rows and columns). - Its determinant is not zero. - Product with its inverse yields the identity matrix: if a matrix \( P \) is invertible, then there exists a matrix \( P^{-1} \) such that \( P P^{-1} = I \), where \( I \) is the identity matrix.The ability to invert a matrix is critical in solving systems of linear equations, finding matrix determinants, and in matrix decomposition. In our exercise, you note matrix \( P \) being invertible ensures that transformations involving \( P \) do not lose any information about the spaces they work with, particularly when considering eigenvectors and eigenspaces.

Eigenspaces

Eigenspaces are collections of all vectors that can be scaled by a matrix without changing direction. Each eigenspace is linked to a specific eigenvalue. If a vector \( \mathbf{x} \) is scaled by a matrix \( A \) to become \( \lambda \mathbf{x} \), then the vector is an eigenvector, and \( \lambda \) is its eigenvalue. The eigenspace for a given \( \lambda \) is the set of all eigenvectors with that eigenvalue, plus the zero vector.- **Understanding Eigenspaces**: - They represent directions in which a matrix stretches or compresses vectors. - They provide insight into the matrix's structural properties and are foundational in applications such as stability analysis and quantum mechanics.In our context, the concept of eigenspaces is extended by matrix transformations like \( P A P^{-1} \). This transformation preserves the eigenvalues, meaning the eigenvectors of this transformed matrix form are simply transformations of the original eigenvectors, linear mapped via \( P \). This understanding is crucial when determining how matrices represent systems and transformations in higher dimensional spaces.