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Jordan Matrix Decomposition


The Jordan matrix decomposition is the decomposition of a square matrix M into the form

 M=SJS^(-1),
(1)

where M and J are similar matrices, J is a matrix of Jordan canonical form, and S^(-1) is the matrix inverse of S. In other words, M is a similarity transformation of a matrix J in Jordan canonical form. The proof that any square matrix can be brought into Jordan canonical form is rather complicated (Turnbull and Aitken 1932; Faddeeva 1958, p. 49; Halmos 1958, p. 112).

Jordan decomposition is also associated with the matrix equation AX=XB and the special case A=B.

The Jordan matrix decomposition is implemented in the Wolfram Language as JordanDecomposition[m], and returns a list {s, j}. Note that the Wolfram Language takes the Jordan block in the Jordan canonical form to have 1s along the superdiagonal instead of the subdiagonal. For example, a Jordan decomposition of

 M=[2 4 -6  0; 4 6 -3 -4; 0 0  4  0; 0 4 -6  2]
(2)

is given by

S=[1 -1/4 0 1; 0  1/4 3 1; 0  0 2 0; 1  0 0 1]
(3)
J=[2 1 0 0; 0 2 0 0; 0 0 4 0; 0 0 0 6].
(4)

See also

Jordan Canonical Form, Matrix Decomposition, Similar Matrices

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References

Faddeeva, V. N. "The Jordan Canonical Form." §4 in Computational Methods of Linear Algebra. New York: Dover, pp. 49-54 and 235, 1958.Frazer, R. A.; Duncan, W. J.; and Collar, A. R. "Collinearity Transformation of a Numerical Matrix to a Canonical Form." §3.16 in Elementary Matrices and Some Applications to Dynamics and Differential Equations. Cambridge, England: Cambridge University Press, pp. 93-95, 1955.Golub, G. H. and Van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, p. 317, 1996.Halmos, P. R. Finite-Dimensional Vector Spaces, 2nd ed. Princeton, NJ: Van Nostrand, p. 112, 1958.Turnbull, H. W. and Aitken, A. C. Chs. 5-6 in An Introduction to the Theory of Canonical Matrices. London: Blackie and Sons, 1932.

Referenced on Wolfram|Alpha

Jordan Matrix Decomposition

Cite this as:

Weisstein, Eric W. "Jordan Matrix Decomposition." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/JordanMatrixDecomposition.html

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