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Math Help - Orthogonal basis

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    Orthogonal basis

    Why is an orthogonal basis important?
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    Re: Orthogonal basis

    Because it is particularly easy to write any vector in terms of an orthogonal basis (even easier if it's orthonormal). Here's an application: thinking of the vector space of functions, you can solve differential equations formed by means of, say, an Hermitian operator, by writing down the solution as a sum of eigenvectors of the operator. Physicists do this kind of thing all the time to solve ODE's or even PDE's.

    Another application is diagonalization. Suppose you have the first-order system of DE's: \dot{\mathbf{x}} = A\mathbf{x}. A solution is e^{At}\mathbf{x}_{0}. But how do you compute e^{At}? Well, if A is diagonalizable (you can find an orthogonal matrix P such that A=PDP^{-1} for some diagonal matrix D), then e^{At}=Pe^{Dt}P^{-1}, which is much easier to compute. And the process of diagonalization, or finding the orthogonal matrix P, is a process of finding an orthonormal (which is automatically orthogonal) basis.
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    Re: Orthogonal basis

    If a basis, \{v_1, v_2, \cdot\cdot\cdot v_n\} is orthonormal (all vectors are orthogonal and have length 1), then the vector v can be written as v= <v,v_1>v_1+ <v,v_2>v_2+\cdot\cdot\cdot+ <v, v_n>v_n. That is, the coefficients of the v_i vector is just the inner product of v with v_i.

    If the basis is orthogonal but the vectors are not of unit length, the coefficient is \frac{<v, v_i>}{<v_i, v_i>}.
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Orthogonal basis

    Also, if B=\{v_1,\ldots,v_n\} is an orthogonal basis of an euclidean space V and x=x_1v_1+\ldots+x_nv_n , y=y_1v_1+\ldots+y_nv_n then, the inner product can be expressed by <x,y>=d_1x_1y_1+\ldots+d_nx_ny_n . If B is an orthonormal basis, <x,y>=x_1y_1+\ldots+x_ny_n that is, the inner product works as in the usual case.

    By the Gram-Schmidt Theorem always is possible to find an orthonormal basis of V . This means that (technically but nor conceptually) things work as if "there is only one inner product" in V .
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