Mathematics/Linear Algebra

주어진 행렬을 LU decomposition을 한 뒤 $L$과 $U$ 행렬 각각의 inverse를 구하면 역행렬을 빠르게 구할 수 있다. 이때 triangular matrix의 inverse를 빠르게 계산하는 방법을 소개하려고 한다. 예컨대 다음과 같은 상삼각 행렬의 역행렬을 계산해 보자. $$U = \begin{pmatrix} 3 & 6 & 8 \\ 0 & 4 & 7 \\ 0 & 0 & 5 \end{pmatrix}$$  이때 각각의 성분에 대해서 따로따로 생각해 보자. Gauss elimination을 생각하면 역행렬의 대각 성분은 원행렬의 대각 성분의 역수가 된다. $$U^{-1} = \begin{pmatrix} \frac{1}{3} & * & * \\ 0 & \frac{1}{4} & * \\ 0 & ..

Positive Definite, Semidefinite Definition 1. Let $T \in \mathcal{L}(V)$ where $V$ is a finite-dimensional inner product space, and let $A \in M_{n \times n}(F)$. Then $T$ is called positive definite [positive semidefinite] if $T$ is hermitian and $\langle T(x), x \rangle > 0$ $[\langle T(x), x \rangle \geq 0], \forall x \neq \mathbf{0}$, and $A$ is called positive definite [positive semidefinit..

Lagrange Interpolation FormulaLagrange Interpolation Formula, 즉 라그랑주 보간법이라고 불리는 이 방법은 주어진 $(n+1)$개의 점들을 모두 지나는 $n$차 이하의 다항식을 유일하게 결정하는 방법이다. Lagrange PolynomialDefinition 1. Let $c_0, \cdots, c_n$ be distinct scalars in an infinite field $F$. The lagrange polynomials $f_0, \cdots, f_n$ is defined by $$f_i(x) = \prod_{0 \leq k \neq i \leq n} \frac{x - c_k}{c_i - c_k} \text{ for } 0 \leq i \leq n...

Spectral Theorem Theorem 1. Let $T \in \mathcal{L}(V)$ where $V$ is a finite-dimensional inner product space over $F$ with the distinct eigenvalues $\lambda_1, \cdots, \lambda_k$. Assume that $T$ is normal if $F = \mathbb{C}$ and that $T$ is hermitian if $F = \mathbb{R}$. For each $i (1 \leq i \leq k)$, let $W_i$ be the eigenspace of $T$ corresponding to the eigenvalue $\lambda_i$, and that $T_i..

Orthogonal Projection Definition 1. Let $T \in \mathcal{L}(V)$ be a projection, where $V$ is an inner product space. We say that $T$ is an orthogonal projection if $R(T)^{\perp} = N(T)$ and $N(T)^{\perp} = R(T)$. Remark Remark. (a) If $V$ is finite-dimensional, by Theorem 2, we need only assume that one of the preceding conditions holds. (b) If $W \leq V$ is finite-dimensional, then there exists..

Projection Definition 1. Let $W_1, \cdots, W_k \leq V$ such that $V = \bigoplus_{i=1}^k W_i$, and let $T \in \mathcal{L}(V)$ where $V$ is a vector space. Then $T$ is the projection on $W_j$ if, whenever $x = x_1 + \cdots + x_k$ with $x_i \in W_i (i = 1, \cdots, k)$, we have $T(x) = x_j$. Theorem 1 Theorem 1. Let $W_1, \cdots, W_k \leq V$ such that $V = \bigoplus_{i=1}^k W_i$ where $V$ is a vecto..

Hermitian MatrixDefintion 1. Let $A \in M_{n \times n}(F)$. We say that $A$ is hermitian (or self-adjoint) if $A = A^*$.선형 연산자가 hermitian일 조건과 동일하게 hermitian인 행렬을 정의할 수 있다. 또한 Theorem 1의 행렬 버전을 말할 수 있다.Theorem 1Theorem 1. Let $A \in M_{n \times n}(\mathbb{R})$. Then $A$ is hermitian $\iff$ $A$ is orthogonally equivalent to a real diagonal matrix. Proof. The proof is similar to the proof of Theor..

Normal MatrixDefintion 1. Let $A \in M_{n \times n}(F)$. We say that $A$ is normal if $AA^* = A^*A$.선형 연산자 $T$가 normal일 조건과 동일하게 normal인 행렬을 정의할 수 있다. 또한 Theorem 2의 행렬 버전을 말할 수 있다.Theorem 1Theorem 1. Let $A \in M_{n \times n}(\mathbb{C})$. Then $A$ is normal $\iff$ $A$ is unitarily equivalent to a diagonal matrix.Proof. ($\Longrightarrow$)By Theorem 2, there exists an orthonormal basis for $V$ c..

Unitary, Orthogonal MatrixDefinition 1. Let $A \in M_{n \times n}(F)$. Then $A$ is called a unitary matrix if $A^*A = AA^* = I$ and is called an orthogonal matrix if $A^tA = AA^t = I$.Theorem 1에 근거해 unitary 혹은 orthogonal 행렬의 정의를 위와 같이 할 수 있다.RemarkRemark. $AA^* = I$ [$A^*A = I$] $\iff$ the rows [columns] of $A$ form an orthonormal basis for $F^n$.($\because$) $\delta_{ij} = I_{Ij} = (AA^*)_{ij} ..

Unitarily, Orthogonally Equivalent Definition 1. Let $A, B \in M_{n \times n}(\mathbb{C})$ [$M_{n \times n}(\mathbb{R})$]. Then $A$ and $B$ are unitarily equivalent [orthogonally equivalent] if there exists a unitary [orthogonal] matrix $P$ such that $A = P^*BP$ [$A = P^tBP$].