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...]

## The Conjugate Transpose of a Matrix

FoldUnfold Table of Contents The Conjugate Transpose of a Matrix The Conjugate Transpose of a Matrix We are about to look at an important theorem which will give us a relationship between a matrix that represents the linear transformation $T$ and a matrix that represents the adjoint of $T$, $T^*$. Before we look at this though, we will need to get a brief definition out … [Read more...]

## Eigenvalues of the Adjoint of a Linear Map

FoldUnfold Table of Contents Eigenvalues of the Adjoint of a Linear Map Eigenvalues of the Adjoint of a Linear Map In the following proposition we will see that the eigenvalues of $T^*$ are the complex conjugate eigenvalues of $T$. Proposition 1: Let $V$ be a finite-dimensional nonzero inner product spaces. Then $\lambda$ is an eigenvalue of $T$ if and only if … [Read more...]

## Injectivity and Surjectivity of the Adjoint of a Linear Map

FoldUnfold Table of Contents Injectivity and Surjectivity of the Adjoint of a Linear Map Injectivity and Surjectivity of the Adjoint of a Linear Map In the following two propositions we will see the connection between a linear map $T$ being injective/surjective and the corresponding adjoint matrix $T^*$ being surjective/injective. Proposition 1: Let $V$ and $W$ be … [Read more...]

## The Null Space and Range of the Adjoint of a Linear Map

FoldUnfold Table of Contents The Null Space and Range of the Adjoint of a Linear Map The Null Space and Range of the Adjoint of a Linear Map Recall from The Adjoint of a Linear Map page that if $V$ and $W$ are finite-dimensional non-zero vector spaces and if $T \in \mathcal L (V, W)$ then the adjoint of $T$ denoted $T^*$ is defined by considering the linear functional … [Read more...]

## Properties of Adjoints of Linear Maps

FoldUnfold Table of Contents Properties of Adjoints of Linear Maps Example 1 Properties of Adjoints of Linear Maps Recall from The Adjoint of a Linear Map page that if $V$ and $W$ are finite-dimensional nonzero inner product spaces and that $T \in \mathcal L (V, W)$ then the adjoint of $T$ is the linear map $T^* \in \mathcal L (W, V)$ defined by considering the linear … [Read more...]