PX153 - K15 - similarity transformation

applying transformation matrix $A$ to $\vec{y} = A \vec{x}$
from previous derivation: $\vec{x} = S {\vec{x}}^{'}$ , $\vec{y} = S {\vec{y}}^{'} :$

\begin{aligned} S {\vec{y}}^{'} & = A S {\vec{x}}^{'} \\ {\vec{y}}^{'} & = S^{- 1} A S {\vec{x}}^{'} \end{aligned}

$S^{- 1} A S$ describes the mapping of matrix $A$ in a new basis of vectors, and this relation, $A \to S^{- 1} A S$ , is called a similarity transformation

properties of similarity transformation

$(i) :$

I^{'} = S^{- 1} I S = I

reminder: for $A^{- 1} A = I$ ,

\begin{aligned} det (A^{- 1} A) = det I & = 1 \\ det A^{- 1} \times det A & = 1 \\ ⟹ det A^{- 1} & = \frac{1}{det A} \end{aligned}

(ii):

\begin{aligned} det A^{'} & = det (S^{- 1} A S) \\ = det S^{- 1} det A det S \\ = \frac{1}{det S} det A det S \\ det A^{'} & = det A \end{aligned}

$(i i i) :$

\begin{aligned} det (A^{'} - λ I^{'}) & = det (S^{- 1} (A - λ I) S) \\ = det S^{- 1} det (A - λ S) det S \\ = det (A - λ I) \end{aligned}

$(i v) :$

T r A^{'} = T r A

- this is a useful as a check for similarity
- proof:

\begin{aligned} T r A^{'} & = \sum_{i} (S^{- 1} A S)_{i i} \\ = \sum_{i, j, k} S_{i j}^{- 1} A_{j k} S_{k i} \\ = \sum_{i, j, k} S_{k i} S_{i j}^{- 1} A_{j k} \\ = \sum_{i, j, k} I_{k j} A_{j k} \\ = \sum_{j} (A I)_{j j} \\ = \sum_{j} A_{j j} \\ = T r A \end{aligned}