Definition (Upper-triangular)
A matrix A = [ a i j ] ∈ F n × n A=[a_{ij}]\in\mathbb{F}^{n\times n} A = [ a ij ] ∈ F n × n upper triangular if a i j = 0 a_{ij}=0 a ij = 0 j < i j<i j < i A : = [ a 1 , 1 a 1 , 2 a 1 , 3 ⋯ a 1 , n 0 a 2 , 2 a 2 , 3 ⋯ a 2 , n 0 0 a 3 , 3 ⋯ a 3 , n ⋮ ⋮ ⋮ ⋱ ⋮ 0 0 0 ⋯ a n , n ] A:=
\begin{bmatrix}
a_{1,1} & a_{1,2} & a_{1,3} &\cdots & a_{1,n} \\
0 & a_{2,2} & a_{2,3} & \cdots & a_{2,n} \\
0 & 0 & a_{3,3} & \cdots & a_{3,n} \\\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & a_{n,n}\\
\end{bmatrix} A := a 1 , 1 0 0 ⋮ 0 a 1 , 2 a 2 , 2 0 ⋮ 0 a 1 , 3 a 2 , 3 a 3 , 3 ⋮ 0 ⋯ ⋯ ⋯ ⋱ ⋯ a 1 , n a 2 , n a 3 , n ⋮ a n , n
Proposition (Criterion for Upper Triangular Matrix)
Let V V V finite dimensional vector space with basis B = { v 1 , ⋯ , v n } B=\{v_1,\cdots,v_n\} B = { v 1 , ⋯ , v n } linear map T ∈ L ( V ) T\in\mathscr{L}(V) T ∈ L ( V )
The matrix of T T T B B B Upper-Triangular
T ( v k ) ∈ \mbox s p a n ( v 1 , ⋯ , v k ) T(v_{k})\in\mbox{span}(v_1,\cdots,v_k) T ( v k ) ∈ \mbox s p an ( v 1 , ⋯ , v k ) 1 ≤ k ≤ n 1\le k\le n 1 ≤ k ≤ n \mbox s p a n ( v 1 , ⋯ , v k ) \mbox{span}(v_1,\cdots,v_k) \mbox s p an ( v 1 , ⋯ , v k ) T T T invariant for 1 ≤ k ≤ n 1\le k\le n 1 ≤ k ≤ n
Theorem (Existence of Upper Triangular Matrices on Complex Spaces)
If V V V C \mathbb{C} C Vector Space and we define some Linear Map T ∈ L ( V ) T\in\mathscr{L}(V) T ∈ L ( V ) Finite Basis B = { v 1 , ⋯ , v n } B=\{v_1,\cdots,v_n\} B = { v 1 , ⋯ , v n } matrix of T T T B B B Upper-Triangular .
Proposition (Criterion for Invertibility using Upper Triangular)
Let V V V Finite Dimensional F \mathbb{F} F Vector Space and T ∈ L ( V ) T\in\mathscr{L}(V) T ∈ L ( V ) Finite Basis of V V V [ a i j ] = M ( T ) [a_{ij}]=\mathcal{M}(T) [ a ij ] = M ( T ) Upper-Triangular with respect to this basis, then T T T invertible if and only if a i , i ≠ 0 ∀ i = 1 , ⋯ , n a_{i,i}\not=0 \ \forall i=1,\cdots,n a i , i = 0 ∀ i = 1 , ⋯ , n
Proposition (Eigenvalues are Diagonal Elements of Upper Triangular)
Let V V V Finite Dimensional Vector Space and let T ∈ L ( V ) T\in\mathscr{L}(V) T ∈ L ( V ) Finite Basis such that [ a i , j ] = M ( T ) [a_{i,j}]=\mathcal{M}(T) [ a i , j ] = M ( T ) Upper-Triangular with respect to this basis, then the eigenvalues of T T T a i , i a_{i,i} a i , i