Maths - Matrix Diagonalisation

Some matrices can be transformed to diagonal matrices, that is, a matrix where the terms not on the leading diagonal are zero.

For a symmetrical matrix we can rotate it to get a diagonal matrix, do some operation, then rotate it back to its original coordinates. This rotation matrix is the eigen matrix or the orthonormal basis of [A], in other words:

[D] = [Q]-1 [A] [Q]

where:

Derivation

The length of a vector squared is given by:

|V|² = Vt * V

where

This length will be unchanged if the coordinates are rotated by a matrix [R]. In this case the vector V is replaced by [R]V and the transposed vector Vt is replaced by Vt[R]t (transposing both operands reverses the order) and the unchanged length is therefore:

|V|² = Vt * V = Vt[R]t[R]V

therefore:

[R]t[R] = [1]

where

See quadratic form.

Inertia Tensor

An example of diagonalisation is an inertia tensor.

  1. Find the eigenvalues a by solving 0 = det{[A] - a[1]) for a. The values of a are the principal moments of inertia.
  2. Find the eigenvectors v of A by solving A v = a v for v.
  3. Normalize the eigenvectors.
  4. Form the matrix C whose whose columns consist of the normalized
  5. D = Ct A C is the diagonal matrix of principal moments of inertia.

In principle, you can write down D directly after (1), however, completing (1) to (5) gives a check on your work.

Note: Ct is the transpose of C.

 

For this case where the only off diagonal terms are 12 and 21, you
know it only needs a rotation about axis 3 to diagonalize it. Use a
similarity transformation:
A'JA where A is the 3x3 rotation matrix about z.
Solve to find
j12 = 0 = j11cos²(a) -j22sin²(a) solve for a
j22 = j22cos²(a) - j11sin²(a)
j11 = j11cos²2(a) - j22sin²(a)
I don't know if that's easier.



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