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...]