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If $A$ is a symmetric matrix, it has a full set of orthonormal eigenvectors ($x_1,...,x_n$)
since $A$ can be represented as $A=Q\lambda Q^{T}$ where $Q$ is an orthogonal matrix.
Then any $x$ is a combination $c_1x_1+...+c_nx_n$ so
$Ax=c_1Ax_1+...+c_nAx_n$

In here, if $A$ has a full set of orthonormal eigenvectors ($x_1,...,x_n$)
why any $x$ is a combination $c_1x_1+...+c_nx_n$?
I'm a bit confused here.
$c_1x_1+...+c_nx_n$ is a linear combination,
and if $c_1=...=c_n=0$ is the only solution of $c_1x_1+...+c_nx_n=0$
then they are linearly independent.

I can't connect this two concepts: linear combination and orthonormal.

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1 Answer 1

up vote 1 down vote accepted

If a set of vectors is orthogonal (in particular, orthonormal), then it is linearly independent. Since you have $n$ of them, this means they form a basis. Therefore, any vector can be uniquely written as a linear combination of them.

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