Let r ∈ℝ and let T ∈ℒ(ℂ^2) be the linear map with canonical matrix T=( 1 -1 -1

step 1 Okay, so we have u equal to one half root 2 0 root 2 1 I root 2 negative 1 1 negative 5 root 2 n

Let r ∈ℝ and let T ∈ℒ(ℂ^2) be the linear map with canonical...

Let $r \in \mathbb{R}$ and let $T \in \mathcal{L}\left(\mathbb{C}^{2}\right)$ be the linear map with canonical matrix $$ T=\left(\begin{array}{cc} 1 & -1 \\ -1 & r \end{array}\right) $$ (a) Find the eigenvalues of $T$. (b) Find an orthonormal basis of $\mathbb{C}^{2}$ consisting of eigenvectors of $T$. (c) Find a unitary matrix $U$ such that $U T U^{*}$ is diagonal.

Let $r \in \mathbb{R}$ and let $T \in \mathcal{L}\left(\mathbb{C}^{2}\right)$ be the linear map with canonical matrix
$$
T=\left(\begin{array}{cc}
1 & -1 \\
-1 & r
\end{array}\right)
$$
(a) Find the eigenvalues of $T$.
(b) Find an orthonormal basis of $\mathbb{C}^{2}$ consisting of eigenvectors of $T$.
(c) Find a unitary matrix $U$ such that $U T U^{*}$ is diagonal.

Key Concepts

Unitary Matrix

A unitary matrix is a complex matrix U that satisfies U*U = I, meaning its conjugate transpose is its inverse. Unitary matrices preserve inner products and norms, making them particularly important in quantum mechanics and numerical linear algebra. They are used in forming an orthonormal basis of eigenvectors, which in turn allows for the diagonalization of normal or Hermitian matrices.

Diagonalization

Diagonalization is the process of finding a basis in which the matrix representation of a linear operator is diagonal. This is useful because diagonal matrices are easier to work with, particularly for computing powers of matrices and exponentials. When a matrix is unitarily diagonalizable, it means there exists a unitary matrix U such that U T U* (where U* is the conjugate transpose of U) is diagonal.

Eigenvalues

Eigenvalues are scalars associated with a linear transformation T such that there exists a nonzero vector (an eigenvector) for which T(v) = ?v. They are found by solving the characteristic equation det(T - ?I) = 0. This concept is central in linear algebra because it describes intrinsic properties of the transformation and is used in many applications including stability analysis and transformation simplification.

Eigenvectors

Eigenvectors are nonzero vectors that only get scaled (not rotated) by a linear transformation, meaning they satisfy T(v) = ?v for a given eigenvalue ?. For matrices with special properties, such as Hermitian (or symmetric in the real case) matrices, one can select these eigenvectors to be orthonormal. This basis simplifies many problems in linear algebra and functional analysis.

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