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By Jorge Antezana y Demetrio Stojanoff

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Extra resources for Analisis Matricial

Example text

An ) =  ... . ...  ∈ Mn (C). 0 0 an 3. 1) = Hermitiana, anti-Hermitiana, normal, definda positiva o unitaria. 2. Sea A ∈ Mn (C) con vector de autovalores λ(A) = (λ1 , ... , λn ). Entonces las siguientes condiciones son equivalentes: 1. A es normal. 2. Para todo x ∈ Cn , Ax = A∗ x . 3. A ∼ = D para cierta matriz diagonal D ∈ Mn (C) . 4. A ∼ = diag (λ(A) ). 5. Existe una BON B = {v1 , . . , vn } de Cn tal que Avi = λi vi para todo i ∈ In . 1 Matrices normales 6. A 2 2 n = 26 |λi |2 i=1 Demostraci´ on.

2. Consideremenos en Cn las siguientes normas: n x 1 |xi | = y x ∞ = m´ax |xi | , i∈In i=1 para todo x ∈ Cn , conocidas como las normas 1 e ∞. Como en el Ejemplo anterior, ellas inducen en Mn (C) las siguientes normas matriciales: Dada A ∈ Mn (C), |||A|||∞ = m´ax x Ax ∞ =1 ∞ |||A|||1 = m´ax Ax y x 1 =1 1 . Probar que estas normas pueden calcularse efectivamente mediante las f´ormulas: |||A|||∞ = m´ax Fi (A) i∈In 1 y |||A|||1 = m´ax Ci (A) i∈In 1 . 8) para toda A ∈ Mn (C). 3. Una norma 1. Matricial: si AB ≤ A · en Mn (C) se llama: B 2.

En efecto, para mostrarlo basta recordar que dados B, C ∈ H(n), vale que B ≤ C ⇐⇒ B x , x ≤ C x , x para todo x unitario en Cn . Notaciones: En el resto de esta secci´on usaremos las siguientes convenciones: 1. Las letras M y S denotar´an subespacios de Cn . 2. Dado M ⊆ Cn , escribiremos M1 = {x ∈ M : x = 1} al conjunto de elementos de M de norma uno. 3 (Courant-Fisher). Sea A ∈ H(n) y sea k ∈ In . Entonces, λk (A) = m´ın m´ax Ax, x = dim M=k x∈M1 m´ax m´ın Ax, x . dim S=n−k+1 x∈S1 Demostraci´ on.

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Analisis Matricial by Jorge Antezana y Demetrio Stojanoff


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