Let A and B be similar matrices, so B=S^-1 A S for some...

Let $A$ and $B$ be similar matrices, so $B=S^{-1} A S$ for some nonsingular matrix $S$. (a) Prove that $A$ and $B$ have the same characteristic polynomial: $p_B(\lambda)=p_A(\lambda)$. (b) Explain why similar matrices have the same eigenvalues. (c) Do they have the same eigenvectors? If not, how are their eigenvectors related? (d) Prove that the converse to part (c) is false by showing that $\left(\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right)$ and $\left(\begin{array}{rr}1 & 1 \\ -1 & 3\end{array}\right)$ have the same eigenvalues, but are not similar.

Let $A$ and $B$ be similar matrices, so $B=S^{-1} A S$ for some nonsingular matrix $S$.
(a) Prove that $A$ and $B$ have the same characteristic polynomial: $p_B(\lambda)=p_A(\lambda)$.
(b) Explain why similar matrices have the same eigenvalues. (c) Do they have the same eigenvectors? If not, how are their eigenvectors related? (d) Prove that the converse to part (c) is false by showing that $\left(\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right)$ and $\left(\begin{array}{rr}1 & 1 \\ -1 & 3\end{array}\right)$ have the same eigenvalues, but are not similar.

Key Concepts

Matrix Similarity

Matrix similarity is a relation where two matrices A and B satisfy the equation B = S?¹AS for some invertible matrix S. This concept captures the idea that A and B represent the same linear transformation expressed in different bases; they share many intrinsic properties including determinant, trace, and eigenvalues, though their entries might be quite different.

Characteristic Polynomial

The characteristic polynomial of a matrix is defined as p(?) = det(?I - A). It is a foundational concept in linear algebra used to compute the eigenvalues of the matrix. The characteristic polynomial is invariant under similarity, meaning similar matrices have identical characteristic polynomials.

Eigenvalues

Eigenvalues are scalar values for which there exists a nonzero vector v (the corresponding eigenvector) satisfying the equation Av = ?v. Since eigenvalues are determined as the roots of the characteristic polynomial, matrices that are similar necessarily have the same eigenvalues.

Eigenvectors

Eigenvectors are the nonzero vectors that remain directionally unchanged by the action of a matrix, and they satisfy the equation Av = ?v for eigenvalue ?. In cases of similarity, eigenvectors of one matrix are related to the eigenvectors of the other through the similarity transformation. However, the eigenvectors themselves are generally not identical because they are expressed in different coordinate systems.

Converse of Eigenvalue-Eigenvector Relations

The converse idea examines whether having the same eigenvalues (or related eigenvectors) implies that two matrices are similar. It is important to understand that identical eigenvalues do not guarantee the matrices are similar, as the geometric multiplicities and the structure of the eigenvectors (or Jordan canonical form) may differ. This demonstrates that eigenvalue sharing is a necessary but not sufficient condition for matrix similarity.

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