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# A Matrix and its Eigenvector

The reason we are stressing eigenvectors and their importance is because the action of a matrix $AA$ on one of its eigenvectors $vv$ is

1. extremely easy (and fast) to calculate
$A⁢v=λ⁢v A v λ v$
1
just multiply $vv$ by $λ λ$.
2. easy to interpret: $AA$ just scales $vv$, keeping its direction constant and only altering the vector's length.
If only every vector were an eigenvector of $AA$....

# Using Eigenvectors' Span

Of course, not every vector can be ... BUT ... For certain matrices (including ones with distinct eigenvalues, $λλ$'s), their eigenvectors span $Cn n$, meaning that for any $x∈Cn x n$, we can find $α 1 α 2 α n ∈C α 1 α 2 α n$ such that:

$x= α 1 ⁢ v 1 + α 2 ⁢ v 2 +…+ α n ⁢ v n x α 1 v 1 α 2 v 2 … α n v n$
2
Given Equation 2, we can rewrite $A⁢x=b A x b$. This equation is modeled in our LTI system pictured below:

$x=∑i α i ⁢ v i x i α i v i$ $b=∑i α i ⁢ λ i ⁢ v i b i α i λ i v i$ The LTI system above represents our Equation 1. Below is an illustration of the steps taken to go from $xx$ to $bb$. $x→(α=V-1⁢x)→(Λ⁢V-1⁢x)→(V⁢Λ⁢V-1⁢x=b) x α V -1 x Λ V -1 x V Λ V -1 x b$ where the three steps (arrows) in the above illustration represent the following three operations:

1. Transform $x x$ using $V-1 V -1$ - yields $αα$
2. Action of $AA$ in new basis - a multiplication by $Λ Λ$
3. Translate back to old basis - inverse transform using a multiplication by $V V$, which gives us $bb$