eigenvectors
Computes eigenvalues and eigenvectors of a matrix.
Syntax
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[Λ, U ≔ ]eigenvectors(A)-
Ais a square matrix
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Description
If A is a square matrix, then eigenvectors(A) tries to compute eigenvalues and eigenvectors of A. eigenvectors(A) is an assignment list containing the vector of all eigenvalues and a matrix whose columns are corresponding eigenvectors.
The function tries to make the matrix of eigenvectors orthonormal.
Examples
A non-symmetric matrix
A ≔ ❨❨1, −1, −1, 1❩, ❨−1, 0, 0, 1❩, ❨−1, 0, 2, 0❩, ❨1, 0, 0, 2❩❩
⎛ 1 −1 −1 1⎞ ⎜−1 0 0 1⎟ ⎜−1 0 2 0⎟ ⎝ 1 0 0 2⎠
Λ, U ≔ eigenvectors(A)
⎛3 ⎞ ⎛ 0.57735026919 0 0.57735026919 0.377964473009⎞ ⎜2 ⎟ ⎜ 0 0.408248290464 0.76980035892 −0.755928946018⎟ e⎜−1⎟ ⎜ −0.57735026919 0.408248290464 0.19245008973 0.377964473009⎟ ⎝1 ⎠ ⎝ 0.57735026919 0.816496580928 −0.19245008973 −0.377964473009⎠
defuzz(U ⋅ diag(Λ) ⋅ U^−1)
⎛ 1 −1 −1 1⎞ ⎜−1 0 0 1⎟ ⎜−1 0 2 0⎟ ⎝ 1 0 0 2⎠
A symmetric matrix
A ≔ ❨❨2, −2, 6, 1❩, ❨−2, 2, 0, 1❩, ❨6, 0, 2, 2❩, ❨1, 1, 2, 2❩❩
⎛ 2 −2 6 1⎞ ⎜−2 2 0 1⎟ ⎜ 6 0 2 2⎟ ⎝ 1 1 2 2⎠
Λ, U ≔ eigenvectors(A)
⎛8.84007170783 ⎞ ⎛ 0.673132037203 0.697804105019 0.181712762859 0.164143727644⎞ ⎜−4.35419040647⎟ ⎜−0.157160718108 0.208318086983 −0.757894177948 0.597913453898⎟ e⎜3.29323039947 ⎟ ⎜ 0.669779412365 −0.681543232662 −0.110680937035 0.273210707991⎟ ⎝0.22088829918 ⎠ ⎝ 0.271273492894 0.071915420233 −0.616706264784 −0.73546736662⎠
IsOrthogonal(U)
true
defuzz(U ⋅ diag(Λ) ⋅ U*)
⎛ 2 −2 6 1⎞ ⎜−2 2 0 1⎟ ⎜ 6 0 2 2⎟ ⎝ 1 1 2 2⎠