# The Mathematics

## The Implicit Function Theorem For Functions from Rn to Rn Examples 1 – The Math

Recall from The Implicit Function Theorem for Functions from Rn to Rn page that if \$A subseteq mathbb{R}^{n+k}\$ is open and \$mathbf{f} : A to mathbb{R}^n\$ is a continuously differentiable function on \$A\$ (\$mathbf{f}\$ is \$C^1\$ on \$A\$) then if there exists an \$(mathbf{x}_0; mathbf{t}_0) in A\$ for which \$mathbf{f}(mathbf{x}_0; mathbf{t}_0) = 0\$ and the …

## The Inclusion Chart for Special Types of Integral Domains – The Math

Below is an inclusion chart showing the various inclusions of the algebraic structures of: commutative rings, integral domains, unique factorization domains, principal ideal domains, Euclidean domains, and Fields: (1) begin{align} quad mathrm{Commutative : Rings} supset mathrm{Integral : Domains} supset mathrm{UFDs} supset mathrm{PIDs} supset mathrm{EDs} supset mathrm{Fields} end{align} The math online

## Subgroups of Cyclic Groups are Cyclic Groups – The Math

Recall from the Cyclic Groups page that a group \$G\$ is said to be a cyclic group if \$G = langle a rangle = { a^n : n in mathbb{Z} }\$ for some \$a in G\$, i.e., \$G\$ can be generated by a single element \$a in G\$. We will now prove an important theorem …

## 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 \$varphi: V to mathbb{F}\$ defined by \$varphi (v) = \$ for a fixed \$w …

## 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 finite-dimensional nonzero inner product spaces and let \$T in mathcal L (V, W)\$. Then \$T\$ is injective if and only if \$T^*\$ is …

## 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 \$overline{lambda}\$ is an eigenvalue of \$T^*\$. Proof: We will prove the contrapositives to proposition …

## 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 of the way in defining a …

## 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 finite-dimensional nonzero inner product spaces. Let …

## Self-Adjoint Linear Operators – The Math

fT[[toc]] 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 …

## 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 important theorem regarding self-adjoint linear operators which says that …