linear algebra proof thanks Let TE L(V) be self-adjoint. Let A₁, A2,..., Am be the distinct eigen-values of T. For i = 1, 2, ..., m, let E; := E(\¡, T) be the corresponding eigenspace,and let PrÅ¡ € L(V) denote the orthogonal projection onto E¿.a.) Prove that for any 1 ≤ i, j≤m,b.) Prove thatPre; Pre; = dij PrEi ·T = A₁ PrE₁ +λ₂ PrE₂ + + Am PremEquation (1) is called the spectral decomposition of T.(1)

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Answered: Let TEL(V) be self-adjoint. Let A₁,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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linear algebra proof

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Let TE L(V) be self-adjoint. Let A₁, A2,..., Am be the distinct eigen-
values of T. For i = 1, 2, ..., m, let E; := E(\¡, T) be the corresponding eigenspace,
and let PrÅ¡ € L(V) denote the orthogonal projection onto E¿.
a.) Prove that for any 1 ≤ i, j≤m,
b.) Prove that
Pre; Pre; = dij PrEi ·
T = A₁ PrE₁ +λ₂ PrE₂ + + Am Prem
Equation (1) is called the spectral decomposition of T.
(1)

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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