Orthogonal Definition 1. Let $(V, \langle \cdot, \cdot \rangle)$ be an inner product space. Let $x, y \in V$, and let $S \subseteq V$. Then (a) $x$ and $y$ are orthogonal (or perpendicular) if $\langle x, y\rangle = 0$. (b) $S$ is orthogonal if any two distinct vector in $S$ are orthogonal. 고등학교 시절 내적을 배웠다면, 내적의 정의를 $x \cdot y = |x| |y| \cos \theta$로 기억하고 있을 것이다. 이 경우 $\theta = 90^{\circ}$일 때 두 ..
Norm Definition 1. Let $V$ be a vector space over $F = \mathbb{R}$ or $\mathbb{C}$. A norm is a function $|| \cdot ||: V \longrightarrow \mathbb{R}$ such that $\forall x, y \in V, \forall a \in F$, the following hold: (a) $||x|| \geq 0$, and $||x|| = 0 \iff x = \mathbf{0}$. (b) $||ax|| = |a|\,||x||$. (c) $||x+y|| \leq ||x|| + ||y||$. Then $(V, ||\cdot||)$ is called a normed space. 복소수의 크기를 절댓값을 ..
Adjoint of Matrix Definition 1. Let $A \in M_{m \times n}(F)$. We define the adjoint or conjugate transpose of $A$ to be the $n \times m$ matrix $A^*$ such that $(A^*)_{ij} = \overline{A_{ji}}$ for all $i, j$. Theorem 1 Theorem 1. Let $A, B \in M_{m \times n}(F)$, and let $C \in M_{n \times p}$. Then (a) $(A+B)^* = A^* + B^*$ (b) $(cA)^* = \overline{c} A^*, \forall c \in F$. (c) $(AC)^* = C^*A^*..
이 포스트에서 $V$는 $F$-벡터공간으로 취급한다. Inner Product Definition 1. An inner product on $V$ is a function $\langle \cdot, \cdot \rangle: V \times V \longrightarrow F$, such that $\forall x, y, z \in V$ and $\forall c \in F$, the following hold: (a) $\langle x + z, y \rangle = \langle x, y \rangle + \langle z, y \rangle$. (b) $\langle cx, y \rangle = c \langle x, y \rangle$. (c) $\overline{ \langle x, y \r..
The Cayley-Hamilton Theorem Theorem 1. (The Cayley-Hamilton Theorem) Let $T \in \mathcal{L}(V)$, and let $f(t)$ be the characteristic polynomial of $T$. (V is finite-dimensional) Then $f(T) = T_0$, the zero transformation. Proof. We need to show that $f(T)(v) = \mathbf{0}, \forall v \in V$. If $v = \mathbf{0}$, it is clear. Suppose that $v \neq \mathbf{0}$. Let $W$ be the $T$-cyclic subspace of ..
