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 Eigenvalues and eigenvectors ▲
Eigenvalues and eigenvectors The system   (λI − A) = 0   has a non trivial solution if and only if det |λI − A| = 0.   If P  denote the matrix of the eigenvectors and   B   denot the diagonal matrix with diagonal elements being the eigenvalues   λi   of   A.
 We can write: AP = PB P − 1 AP = B         or         A = PBP − 1
In this case,   A   is said to be similar to   B. Example:   Find the eigenvalues and the eigenvectors of the       matrix A = From the eigenvalues equation we get the characteristic polinomial:   The eigenvalues are the roots of the characteristic polinomial, and are: − 2, − 2,   4   this values are the diagonal values and has the same determinant value as matrix     A.  In order to find the eigenvectors substitute first solution λ =− 2   into the characteristic matrix,the result is: Those equations reduces to one non independed equation: x − y + z = 0 Choose arbitrary   y = 0   to recieve the first eigenvector x = 1,       y = 0,       z = − 1 Choose arbitrary   z = 0   to get the second vector: x = 1,       y = 1,       z = 0
Perform the same process with the third solution   λ = 4   to get: The result is two independed equations: x + y − z = 0         and         2y − z = 0 Choose arbitrary   y = 1   to recieve the vector x = 1,         y = 1,         z = 2
 The three eigenvectors are: We can see that the result is matching to the definition of the eigenvectors and eigenvalues: Rotation Matrices ▲
Rotation of a vector is performed by applying the rotation matrix   R on the vector   V.           V' = R × V If the rows and columns of a rotation matrix R are orthogonal to each other and of unit length then the following relations are true:
RT = R− 1             RT R = RRT = I             detR = 1
Two dimentional rotation
Rotation by θ counterclockwise +θ counterclockwise
−θ clockwise
Rotation by θ clockwise sin(−θ) = −sinθ
cos(−θ) = cosθ
Rotation by 90° counterclockwise Rotation by 180° counterclockwise Rotation by 270° counterclockwise Three dimentional rotation (θ − x axis,       ϕ − y axis,       φ − z axis)
 Rotation aboutx axis   Rotation about>y axis   Rotation aboutz axis    Combined rotation in the direction of all axes can be done by multiplying the three rotation matrices in the x,y and z direction
 Counterclockwise: Rxyz = Rx(θ)∙Ry(ϕ)∙Rz(φ) Mixed direction: Rxyz = Rx(−θ)∙Ry(−ϕ)∙Rz(φ)
Rotation in the direction of two axes can be done by:
 Rxy = Rx Ry or Rxz = Rx Rz or Ryz = Ry Rz
clockwise rotation matrix of the axes (vector is moved counterclockwise)
Note: the order of the rotation is important as rotation Rx (θ) Ry (ϕ) Rz (φ)   is not equal to rotation
Rz (φ) Ry (ϕ) Rx (θ). Counterclockwise rotation matrix of the axes (vector is moved clockwise) Example:
Rotate the vector V = 2i + j
 by 30° counterclockwise. (Rotated vector) V ' = R × V           (R - Rotation matrix) V' = [2 cos(30°) − sin(30°)]i + [2 sin(30°) + cos(30°)]j
V' = 1.23i + 1.87j
Example:
Rotate the vector   V = i + j + k   by an angle of   30°   counterclockwise about the x axis,   45°   clockwise about the   y   axis and   60°   clockwise about   z   axis.
Step 1: rotation θ = 30° about x axis counterclockwise: Step 2: rotation ϕ = 45° about y axis clockwise: Step 3: rotation φ = 60° about z axis clockwise: 