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Jacobi Rotation Matrix

A matrix used in the Jacobi transformation method of diagonalizing matrices. The Jacobi rotation matrix P_(pq) contains 1s along the diagonal, except for the two elements cosphi in rows and columns p and q. In addition, all off-diagonal elements are zero except the elements sinphi and -sinphi. The rotation angle phi for an initial matrix A is chosen such that

cot(2phi)=(a_(qq)-a_(pp))/(2a_(pq)).

Then the corresponding Jacobi rotation matrix which annihilates the off-diagonal element a_(pq) is

P_(pq)=[1 0; ... | ... ; cosphi ... 0 ... sinphi ; ... 0 ... 1 ... 0 ... ; -sinphi ... 0 ... cosphi ; ... | ... ; 0 1]

See also

Jacobi Transformation

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References

Gentle, J. E. "Givens Transformations (Rotations)." §3.2.5 in Numerical Linear Algebra for Applications in Statistics. Berlin: Springer-Verlag, pp. 99-102, 1998.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Jacobi Transformation of a Symmetric Matrix." §11.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 456-462, 1992.

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Jacobi Rotation Matrix

Cite this as:

Weisstein, Eric W. "Jacobi Rotation Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JacobiRotationMatrix.html

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