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 ProductDefinition 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 \rangle..

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 ..

The Cyclic SubspaceDefinition 1. Let $T \in \mathcal{L}(V)$, and let a nonzero vector $x \in V$. The subspace $W = \langle x, T(x), T^2(x), ... \rangle$ is called the $T$-cyclic subspace of $V$ generated by $x$.Theorem 1Theorem 1. Let $T \in \mathcal{L}(V)$, and let $W$ be the $T$-cyclic subspace of $V$ generated by $\mathbf{0} \neq x \in V$. Then(a) $W$ is $T$-invariant.(b) Any $T$-invariant su..

The Invariant SubspaceDefinition 1. Let $T \in \mathcal{L}(V)$. Then $W \leq V$ is called a $T$-invariant subspace of $V$ if $T(W) \subseteq W$.    $W$의 image가 다시 $W$에 포함될 때 $W$를 $T$-불변 부분공간이라고 부른다. 자명하게 $\{\mathbf{0}\}, V, R(T), N(T), E_{\lambda}$는 $T$-불변 부분공간임을 알 수 있다.The restriction of a Linear OperatorDefinition 2. Let $T \in \mathcal{L}(V)$, and let $W$ be a $T$-invariant subspace of $V$. T..

어떤 선형 연산자 $T$가 주어졌을 때 대각화가능한지 결정하고, 가능하다면 대각화하도록 고유벡터들로 이루어진 기저 $\beta$를 찾는 것이 우리의 목표이다. $T$의 고유값은 특성 다항식 $f(t) = \det (T - tI)$를 풀어서 구할 수 있다. 만약 이를 통해 서로 다른 고유값 $\lambda_1, ..., \lambda_k$를 구했을 때, 이 고유값들에 대응되는 고유벡터들은 $v \in E_{\lambda}$을 이용해서 구할 수 있다. 이제 이 고유벡터들로 기저를 구성해야 하고, 그 방법을 아래의 정리들이 제시해준다. Theorem 1Theorem 1. Let $T \in \mathcal{L}(V)$, and let $\lambda_1, ..., \lambda_k$ be distinct ei..

The MultiplicityDefintion 1. Let $T \in \mathcal{L}(V)$, and let $\lambda$ be an eigenvalue of $T$ with characteristic polynomial $f(t)$. Then (a) The algebric multiplicity of $\lambda$ is the largest positive integer $k$ for which $(t - \lambda)^k$ is a factor of $f(t)$.(b) The geometric multiplicity of $\lambda$ is $\dim(E_{\lambda})$ where $E_{\lambda}$ is the eigenspace of T corresponding to..

The EigenspaceDefinition 1. Let $T \in \mathcal{L}(V)$, and let $\lambda$ be an eigenvalue of $T$. The eigenspace of $T$ corresponding to $\lambda$ is the set $E_{\lambda} = N(T - \lambda I_V) = \{x \in V \,|\, T(x) = \lambda x\}$. Analogously, we define the eigenspace of a square matrix $A$ to be the eigenspace of $L_A$.    즉 주어진 고유벡터 $\lambda$에 대응하는 고유공간 $E_{\lambda}$는 $\lambda$에 대응하는 고유벡터들과 영..

이 포스트에서 $V$는 $n$차원 $F$-벡터공간으로 취급한다.Theorem 1Theorem 1. (a) Let $T \in \mathcal{L}(V)$. Then a scalar $\lambda$ is an eigenvalue of $T$ $\Longleftrightarrow$ $\det(T - \lambda I_V) = 0$.(b) Let $A \in M_{n \times n}(F)$. Then a scalar $\lambda$ is an eigenvalue of $A$ $\Longleftrightarrow$ $\det(A - \lambda I_n) = 0$.Proof. (a) Since $\lambda$ is an eigenvector of $T$, there is an nonzero vector ..

이 포스트에서 $V$는 유한차원 $F$-벡터공간으로 취급한다.DiagonalizableDefinition 1. Let $T \in \mathcal{L}(V) [A \in M_{n \times n}(F)]$. $T [A]$ is called diagonalizable if there is an ordered basis $\beta$ for $V [F^n]$ such that $[T]_{\beta} [[L_A]_{\beta}]$ is a diagonal matrix.Eigenvector, EigenvalueDefinition 2. Let $T \in \mathcal{L}(V) [A \in M_{n \times n}(F)]$. Then $\mathbb{0} \neq v \in V [F^n]$ is called..