Consider a linear transformation $\mathbf{T}: \mathbf{V} \to \mathbf{V}$ on a finite-dimensional vector space $\mathbf{V}$. If $\mathbf{A}$ is the matrix representation of $\mathbf{T}$ with respect to some basis, which of the following statements is true regarding the eigenvalues of $\mathbf{T}$ and $\mathbf{A}$?
Question 2
Given a matrix $\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, calculate its eigenvalues.
Question 3
If a matrix $\mathbf{A}$ has a repeated eigenvalue $\lambda$ with algebraic multiplicity $m > 1$, what condition must be met for $\mathbf{A}$ to be diagonalizable?
Question 4
What is the primary significance of the spectral theorem for symmetric matrices?
Question 5
Given a matrix $\mathbf{A} = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$, find the geometric multiplicity of its eigenvalue.
