strangerRidingCaml

Spectral TheoryHermitian, Unitary, and Normal MatricesA matrix $\mathbf{A}$ is: Hermitian if it is equal to its conjugate transpose: $\mathbf{A} = \mathbf{A}^*$. Unitary if its conjugate transpose is its inverse: $\mathbf{A}^* \mathbf{A} = \mathbf{A} \mathbf{A}^* = \mathbf{I}$. Normal if it commutes with its conjugate transpose: $\mathbf{A} \mathbf{A}^* = \mathbf{A}^* \mathbf{A}$.Spectr..

Inner Product SpacesInner Product, Norms, and Orthogonality in Euclidean SpacesAn inner product on a vector space $V$ is a function $\langle \cdot, \cdot \rangle: V \times V \rightarrow \mathbb{R}$ that satisfies the following properties for all vectors $\mathbf{u}, \mathbf{v}, \mathbf{w}$ and scalars $a, b$: Linearity: $\langle a\mathbf{u} + b\mathbf{v}, \mathbf{w} \rangle = a\langle \mathbf..

Eigenvalues and EigenvectorsDefinition and Characteristic EquationLet $\mathbf{A}$ be an $n \times n$ matrix. A scalar $\lambda$ is called an eigenvalue of $\mathbf{A}$ if there exists a non-zero vector $\mathbf{v}$ such that: $$\mathbf{Av} = \lambda \mathbf{v}$$ The characteristic equation of $\mathbf{A}$ is given by: $$\text{det}(\mathbf{A} - \lambda \mathbf{I}) = 0$$ where $\mathbf{I}$ is the ide..

Linear TransformationsDefinition and PropertiesA linear transformation $T: \mathbb{R}^m \rightarrow \mathbb{R}^n$ is a function that preserves vector addition and scalar multiplication:$T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ for all vectors $\mathbf{u}, \mathbf{v}$ in the domain of $T$.$T(c\mathbf{v}) = cT(\mathbf{v})$ for all scalar $c$ and vector $\mathbf{v}$ in the domain..

Vector SpacesBasis and DimensionA basis of a vector space $V$ is a set of linearly independent vectors that span $V$. The dimension of $V$, denoted as $\text{dim}(V)$, is the number of vectors in any basis of $V$.Orthogonality, Orthogonal Complements, and ProjectionsTwo vectors $\mathbf{v}$ and $\mathbf{w}$ in a vector space are orthogonal if their dot product is zero, i.e., $\mathbf{v} \cdot \m..

Systems of Linear EquationsGaussian Elimination and Row Echelon FormGaussian elimination is a method used to solve systems of linear equations by performing row operations on the augmented matrix of the system until it is in row echelon form.Matrix Equations, Existence, and Uniqueness of SolutionsA system of linear equations can be represented as a matrix equation $\mathbf{Ax} = \mathbf{b}$, whe..