16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 8.5.4

Inner Product Spaces

Mathematics — Linear Algebra · normal-operators-orthonormal-basis

Problem

Let VV be C2C^2, with the standard inner product. Let TT be the linear operator on VV which is represented in the standard ordered basis by the matrix A=[1ii1].A = \begin{bmatrix} 1 & i \\ i & 1 \end{bmatrix}. Show that TT is normal, and find an orthonormal basis for VV, consisting of characteristic vectors for TT.
Intuitive explanation

The key insight is that a matrix is normal when it commutes with its conjugate transpose (AA∗=A∗AAA^* = A^*A), which guarantees it's diagonalizable by a unitary transformation. Once you verify normality, find the eigenvalues and eigenvectors of AA, then normalize them using the standard inner product on C2\mathbb{C}^2—the eigenvectors of a normal operator corresponding to different eigenvalues are automatically orthogonal, so you just need to scale them to unit length.

Intuitive explanation

The key insight is that a matrix is normal when it commutes with its conjugate transpose (AA∗=A∗AAA^* = A^*A), which guarantees it's diagonalizable by a unitary transformation. Once you verify normality, find the eigenvalues and eigenvectors of AA, then normalize them using the standard inner product on C2\mathbb{C}^2—the eigenvectors of a normal operator corresponding to different eigenvalues are automatically orthogonal, so you just need to scale them to unit length.

Related theory

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