Spectral theorem

For an $m\times n$ matrix $A$, its Hermitian adjoint $A^{\ast }$ is defined by $A^{\ast }=\overline{A^T}$. The following identity is the main property of the adjoint matrix:

$$\begin{array}{rlr}⟨A\mathbf{x},\mathbf{y}⟩=⟨\mathbf{x},A^{\ast }\mathbf{y}⟩& & \forall \mathbf{x}\in {\mathbb{C}}^n,\mathbf{y}\in {\mathbb{C}}^m.\end{array}$$

The above identity is often used as the definition of the adjoint operator, and it uniquely defines the adjoint.

An operator $U:X\to Y$ is called an isometry, if it preserves the norm. This is equivalent to preserving the inner product. An operator is an isometry iff $U^{\ast }U=I$. An invertible isometry is called a unitary operator. A square matrix $U$ is called unitary if $U^{\ast }U=I$. A matrix $U$ is an isometry iff its columns form an orthonormal system.

Matrices $A$ and $B$ are called unitary equivalent if there exists a unitary operator $U$ such that $A=UB{U}^{\ast }$.

A Matrix $A$ is unitarily equivalent to a diagonal one iff it has an orthogonal basis of eigenvectors.

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Every complex matrix $A$ is unitary equivalent to an upper triangular matrix.

Every real matrix $A$ is unitary(orthogonal) equivalent to a real upper triangular matrix if all of its eigenvalues are real.

Let $A=A^{\ast }$ be a self-adjoint matrix. Then $A$ can be represented as $A=UD{U}^{\ast }$, where $U$ is a unitary matrix and $D$ is a diagonal matrix with real entries.