Let be a symmetric matrix, and be a Givens rotation matrix. Then: is symmetric and similar to. Furthermore, has entries: where and. Since is orthogonal, and have the same Frobenius norm , however we can choose such that, in which case has a larger sum of squares on the diagonal: Set this equal to 0, and rearrange: if In order to optimize this effect, Sij should be the off-diagonal element with the largest absolute value, called the pivot. The Jacobi eigenvalue method repeatedly performs rotations until the matrix becomes almost diagonal. Then the elements in the diagonal are approximations of the eigenvalues of S.
Convergence
If is a pivot element, then by definition for . Let denote the sum of squares of all off-diagonal entries of. Since has exactly off-diagonal elements, we have or . Now. This implies or , i.e. the sequence of Jacobi rotations converges at least linearly by a factor to a diagonal matrix. A number of Jacobi rotations is called a sweep; let denote the result. The previous estimate yields i.e. the sequence of sweeps converges at least linearly with a factor ≈ . However the following result of Schönhage yields locally quadratic convergence. To this end let S have m distinct eigenvalues with multiplicities and let d > 0 be the smallest distance of two different eigenvalues. Let us call a number of Jacobi rotations a Schönhage-sweep. If denotes the result then Thus convergence becomes quadratic as soon as
Cost
Each Jacobi rotation can be done in O steps when the pivot element p is known. However the search for p requires inspection of all N ≈ ½ n2 off-diagonal elements. We can reduce this to O complexity too if we introduce an additional index array with the property that is the index of the largest element in row i, of the current S. Then the indices of the pivot must be one of the pairs. Also the updating of the index array can be done in O average-case complexity: First, the maximum entry in the updated rows k and l can be found in O steps. In the other rows i, only the entries in columns k and l change. Looping over these rows, if is neither k nor l, it suffices to compare the old maximum at to the new entries and update if necessary. If should be equal to k or l and the corresponding entry decreased during the update, the maximum over row i has to be found from scratch in O complexity. However, this will happen on average only once per rotation. Thus, each rotation has O and one sweep O average-case complexity, which is equivalent to one matrix multiplication. Additionally the must be initialized before the process starts, which can be done in n2 steps. Typically the Jacobi method converges within numerical precision after a small number of sweeps. Note that multiple eigenvalues reduce the number of iterations since.
Algorithm
The following algorithm is a description of the Jacobi method in math-like notation. It calculates a vector e which contains the eigenvalues and a matrix E which contains the corresponding eigenvectors, i.e. is an eigenvalue and the column an orthonormal eigenvector for, i = 1, …, n. procedure jacobi var i, k, l, m, state ∈ N s, c, t, p, y, d, r ∈ R ind ∈ Nn changed ∈ Ln function maxind ∈ N ! index of largest off-diagonal element in row k m := k+1 fori := k+2 tondo if │Ski│ > │Skm│ thenm := iendif endfor returnm endfunc procedure update ! update ek and its status y := ek; ek := y+t ifchangedk and thenchangedk := false; state := state−1 elsif and thenchangedk := true; state := state+1 endif endproc procedure rotate ! perform rotation of Sij, Skl ┌ ┐ ┌ ┐┌ ┐ │Skl│ │c −s││Skl│ │ │ := │ ││ │ │Sij│ │sc││Sij│ └ ┘ └ ┘└ ┘ endproc ! init e, E, and arrays ind, changed E := I; state := n fork := 1 tondoindk := maxind; ek := Skk; changedk := true endfor whilestate≠0 do ! next rotation m := 1 ! find index of pivot p fork := 2 ton−1 do if │Skindk│ > │Smindm│ thenm := kendif endfor k := m; l := indm; p := Skl ! calculate c = cos φ, s = sin φ y := /2; d := │y│+√ r := √; c := d/r; s := p/r; t := p2/d ify<0 thens := −s; t := −tendif Skl := 0.0; update; update ! rotate rows and columns k and l fori := 1 tok−1 do rotate endfor fori := k+1 tol−1 do rotate endfor fori := l+1 tondo rotate endfor ! rotate eigenvectors fori := 1 tondo ┌ ┐ ┌ ┐┌ ┐ │Eik│ │c −s││Eik│ │ │ := │ ││ │ │Eil│ │sc││Eil│ └ ┘ └ ┘└ ┘ endfor ! rows k, l have changed, update rows indk, indl indk := maxind; indl'' := maxind loop endproc
Example
Let Then jacobi produces the following eigenvalues and eigenvectors after 3 sweeps :
When the eigenvalues of a symmetric matrix are known, the following values are easily calculated. ;Singular values ;2-norm and spectral radius ;Condition number ;Rank ;Pseudo-inverse ;Least squares solution ;Matrix exponential ;Linear differential equations
Generalizations
The Jacobi Method has been generalized to complex Hermitian matrices, general nonsymmetric real and complex matrices as well as block matrices. Since singular values of a real matrix are the square roots of the eigenvalues of the symmetric matrix it can also be used for the calculation of these values. For this case, the method is modified in such a way that S must not be explicitly calculated which reduces the danger of round-off errors. Note that with . The Jacobi Method is also well suited for parallelism.
