since i don’t have algebra book on my hand, so, it is just a reminder, very basic thing.

for any matrix M , it can be diagonalized by it eigenvalue \lambda_i  and eigen vector v_i , given that it eigenvectors span all the space. thus, the transform represented by the matrix not contractive, which is to say, the dimension of the transform space is equal to the dimension of the origin space.

Let denote, D before Diagonal matrix, with it elements are eigenvalues.

D_{ij} = \lambda_i \delta_{ij}

P be the matrix that collect the eigenvectors:

P_{i j} = \left( v_i \right)_j = \begin {pmatrix} v_1 & v_2 & ... & v_i \end {pmatrix}

Thus, the matrix M is :

M = P \cdot D \cdot P^{-1}

there are some special case. since any matrix can be rewritten by symmetric matrix S and anti-symmetric matrix A . so we turn our focus on these 2 matrices.

For symmetric matrix S , the transpose of P also work

S =P \cdot D \cdot P^{-1} = (P^T)^{-1} \cdot D \cdot P^T

which indicated that P^T = P^{-1} . it is because, for a symmetric matrix, M = M^T ,  the eigenvalues are all different, then all eigenvector are all orthogonal, thus P^T \cdot P = 1 .

For anti-symmetric matrix A

A = P \cdot D \cdot P^{-1}

since the interchange of row or column with corresponding exchange of eigenvalues in D still keep the formula working. Thus, the case P = P^T never consider.

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For example, the Lorentz Transform

L = \gamma \begin {pmatrix} \beta & 1 \\ 1 & \beta \end {pmatrix}

which has eigenvalues:

D = \gamma \begin {pmatrix} \beta-1 & 0 \\ 0 & \beta+1 \end {pmatrix}

P = \begin {pmatrix} -1 & 1 \\ 1 & 1 \end {pmatrix}

the eigenvector are the light cone. because only light is preserved in the Lorentz Transform.

and it is interesting that

L = P \cdot D \cdot P^{-1} = P^{-1} \cdot D \cdot P = P^T \cdot D \cdot (P^T)^{-1} = (P^T)^{-1} \cdot D \cdot P^T

another example is the Rotation Matrix

R = \begin {pmatrix} cos(\theta) & - sin(\theta) \\ sin(\theta) & cos(\theta) \end{pmatrix}

D = \begin {pmatrix} Exp( - i \theta) & 0 \\ 0 & Exp(i \theta) \end {pmatrix}

P = \begin {pmatrix} -i & i \\ 1 & 1 \end{pmatrix}

the last example to give is the J_x of the spin-½ angular momentum

J_x = \begin {pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

D = \begin {pmatrix} -1 & 0 \\ 0 & 1 \end {pmatrix}

P = \begin {pmatrix} -1 & 1 \\ 1 & 1 \end{pmatrix}

 

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