FoldUnfold Table of Contents Normal Linear Operators Normal Linear Operators We are now going to look at another important type of linear operator known as Normal linear operators which we define below. Definition: Let $V$ be a finite-dimensional inner product space. Then a linear operator $T \in \mathcal L(V)$ is said to be Normal if $TT^* = T^*T$. Linear operators … [Read more...]

## Self-Adjoint Linear Operators over Complex Vector Spaces

FoldUnfold Table of Contents Self-Adjoint Linear Operators over Complex Vector Spaces Self-Adjoint Linear Operators over Complex Vector Spaces Recall from the Self-Adjoint Linear Operators page that if $V$ is a finite-dimensional nonzero inner product space and if $T \in \mathcal L (V)$ then $T$ is said to be self-adjoint if $T = T^*$. In the following proposition we will … [Read more...]

## Eigenvalues of Self-Adjoint Linear Operators

FoldUnfold Table of Contents Eigenvalues of Self-Adjoint Linear Operators Eigenvalues of Self-Adjoint Linear Operators Recall from the Self-Adjoint Linear Operators page that if $V$ is a finite-dimensional nonzero inner product space then $T \in \mathcal L (V)$ is said to be self-adjoint if $T = T^*$ (that is $T$ equals its adjoint $T^*$). We will now look at a very … [Read more...]

## Self-Adjoint Linear Operators

fT[[toc]] Self-Adjoint Linear Operators Recall that if $V$ and $W$ are finite-dimensional nonzero inner product space and if $T \in \mathcal L(V, W)$ them the adjoint of $T$ denoted $T^*$ is the linear map $T^* : W \to V$ is defined by considering the linear function $\varphi : V \to \mathbb{F}$ defined by $\varphi (v) = $ and for a fixed $w \in W$ we define $T^* (w)$ to be the … [Read more...]

## The Matrix of the Adjoint of a Linear Map

FoldUnfold Table of Contents The Matrix of the Adjoint of a Linear Map The Matrix of the Adjoint of a Linear Map Recall from the The Conjugate Transpose of a Matrix page that if $A$ is an $m \times n$ matrix then the conjugate transpose of $A$ is the matrix obtained by taking the complex conjugate of each entry in $A$ and then transposing $A$. Now let $V$ and $W$ be … [Read more...]