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 Formula Lagrange Interpolation Formula, 즉 라그랑주 보간법이라고 불리는 이 방법은 주어진 $(n+1)$개의 점들을 모두 지나는 $n$차 이하의 다항식을 유일하게 결정하는 방법이다. Lagrange Polynomial Definition 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 ..
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..
