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JordanModelDecomposition[ssm]

yields the Jordan decomposition of the state-space model ssm.

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Basic Examples  
Scope  
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Properties & Relations  
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JordanModelDecomposition

JordanModelDecomposition[ssm]

yields the Jordan decomposition of the state-space model ssm.

Details

  • The result is a list {p,jc}, where p is a similarity matrix, and jc is the Jordan canonical form of ssm.
  • The state-space model ssm can be given as StateSpaceModel[{a,b,c,d}], where a, b, c, and d represent the state, input, output, and transmission matrices in either a continuous-time or a discrete-time system:
  • continuous-time system
    discrete-time system
  • The transformation , where is the new state vector, and is a similarity matrix that spans the linearly independent eigenvectors of , transforms the system into the Jordan canonical form:
  • w'(t)=TemplateBox[{p}, Inverse].a.p.w(t)+TemplateBox[{p}, Inverse].b.u(t), continuous-time system
    w(k+1)=TemplateBox[{p}, Inverse].a.p.w(k)+TemplateBox[{p}, Inverse].b.u(k),.discrete-time system
  • The new state matrix TemplateBox[{p}, Inverse].a.p is the Jordan canonical form of the old state matrix .

Examples

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Basic Examples  (1)

Compute the Jordan decomposition of a state-space model:

Wolfram Language code: JordanModelDecomposition[StateSpaceModel[{{{0, 0, -4}, {1, -2, -2}, {-1, 0, 4}}, {{-1}, {0}, {0}}, {{0, 0, 1}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]

Scope  (4)

The Jordan decomposition of a second-order system:

Wolfram Language code: JordanModelDecomposition[StateSpaceModel[{{{-1, 0}, {3, 2}}, {{1}, {-1}}, {{1, 0}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]

The Jordan decomposition of a discrete-time system:

Wolfram Language code: JordanModelDecomposition[StateSpaceModel[{{{0.1625, -0.1125}, {0.1125, 0.5375}}, {{0.5}, {-0.5}}, {{2, -2}}, {{0}}}, SamplingPeriod -> 0.5, SystemsModelLabels -> None]]

A transformation that gives the complex poles in second-order blocks:

Wolfram Language code: Chop[JordanModelDecomposition[StateSpaceModel[TransferFunctionModel[{{{3}}, (-0.2 + z)*(1 + z + z^2)}, z, SamplingPeriod -> 0.1]]]]

Repeated poles appear in Jordan blocks:

Wolfram Language code: JordanModelDecomposition[StateSpaceModel[TransferFunctionModel[{{{10}}, (1 + s)^3*(5 + s)}, s]]]

Applications  (2)

A system is controllable if and only if the Jordan blocks of have distinct eigenvalues, and the row of corresponding to the last row of each Jordan block is not zero:

Wolfram Language code: lssm = {StateSpaceModel[{{{-1, 0}, {0, -2}}, {{1}, {-1}}}, SamplingPeriod -> None, SystemsModelLabels -> None], StateSpaceModel[{{{-1, 1, 0, 0}, {0, -1, 1, 0}, {0, 0, -1, 0}, {0, 0, 0, -2}}, {{1, 0, 0}, {0, 0, 1}, {0, 0, 0}, {0, -1, 0}}}, SamplingPeriod -> None, SystemsModelLabels -> None], StateSpaceModel[{{{-1, 1, 0, 0}, {0, -1, 1, 0}, {0, 0, -1, 0}, {0, 0, 0, -2}}, {{0, 0, 0}, {0, 0, 0}, {1, 0, 0}, {0, -1, 0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]};
Wolfram Language code: Table[ControllableModelQ[ssm], {ssm, lssm}]

A system is observable if and only if the Jordan blocks of have distinct eigenvalues, and the column of corresponding to the first row of each Jordan block is not zero:

Wolfram Language code: lssm = {StateSpaceModel[{{{-1, 0}, {0, -2}}, {{1}, {-1}}, {{2, -2}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None], StateSpaceModel[{{{-1, 1, 0, 0}, {0, -1, 1, 0}, {0, 0, -1, 0}, {0, 0, 0, -2}}, {{1}, {1}, {0}, {-1}}, {{0, 1, 1, 1}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None], StateSpaceModel[{{{-1, 1, 0, 0}, {0, -1, 1, 0}, {0, 0, -1, 0}, {0, 0, 0, -2}}, {{0}, {0}, {1}, {0}}, {{1, 0, 0, 1}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None]};
Wolfram Language code: Table[ObservableModelQ[ssm], {ssm, lssm}]

Properties & Relations  (3)

In the Jordan canonical form, the eigenvalues are along the diagonal of the state matrix:

Wolfram Language code: m = (⁠| | | | | -- | -- | -- | | -1 | 0 | 0 | | 0 | -2 | 0 | | 0 | 0 | -3 |⁠);
Wolfram Language code: p = (⁠| | | | | -- | -- | -- | | 0 | 1 | 0 | | -1 | 0 | -1 | | 1 | -1 | 0 |⁠);
Wolfram Language code: JordanModelDecomposition[StateSpaceModel[{Inverse[p].m.p, (⁠| | | | -- | -- | | 0 | 0 | | b1 | 0 | | 0 | b2 |⁠), (⁠| | | | | -- | - | -- | | c1 | 0 | 0 | | 0 | 0 | c3 |⁠), (⁠| | | | --- | --- | | d11 | d12 | | d21 | d22 |⁠)}]]//Last

The Jordan canonical form is related to the original system via the similarity transform:

Wolfram Language code: ssm = StateSpaceModel[{{{-2, 1, 3}, {1, 0, -1}, {2, 0, 1}}, {{1}, {-1}, {0}}, {{1, 0, 0}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: {p, jc} = N[JordanModelDecomposition[ssm]];
Wolfram Language code: {jc, StateSpaceTransform[ssm, p]//Chop}
Wolfram Language code: {ssm, StateSpaceTransform[jc, Inverse@p]//Chop}

The Jordan canonical form of a state-space model is the similarity transformation associated with the Jordan decomposition of its state matrix:

Wolfram Language code: ssm = StateSpaceModel[{{{-7, 0.5, 5}, {-4, -1, 4}, {-2, 0.5, 0}}, {{1}, {-1}, {1}}, {{1, -1, -1}}, {{0}}}, SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: {p, jc} = JordanDecomposition@First@Normal@ssm
Wolfram Language code: {Last@JordanModelDecomposition[ssm], StateSpaceTransform[ssm, p]}//Chop

Possible Issues  (1)

JordanModelDecomposition does not support descriptor systems:

Wolfram Language code: JordanModelDecomposition[StateSpaceModel[{{{1, 0, 2, -2}, {3, -3, -1, -1}, {2, -1, 0, 1}, {15, -15, 0, 5}}, {{0}, {1}, {0}, {1}}, {{1383, -1481, -16, 403}}, {{0}}, {{3, -3, -1, -1}, {0, 0, 0, 0}, {0, 0, 0, 0}, {-3, 3, 0, -1}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]

Use KroneckerModelDecomposition to separate the modes of the system:

Wolfram Language code: KroneckerModelDecomposition[StateSpaceModel[{{{1, 0, 2, -2}, {3, -3, -1, -1}, {2, -1, 0, 1}, {15, -15, 0, 5}}, {{0}, {1}, {0}, {1}}, {{1383, -1481, -16, 403}}, {{0}}, {{3, -3, -1, -1}, {0, 0, 0, 0}, {0, 0, 0, 0}, {-3, 3, 0, -1}}}, SamplingPeriod -> None, SystemsModelLabels -> None]]
Wolfram Research (2010), JordanModelDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/JordanModelDecomposition.html.

Text

Wolfram Research (2010), JordanModelDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/JordanModelDecomposition.html.

CMS

Wolfram Language. 2010. "JordanModelDecomposition." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/JordanModelDecomposition.html.

APA

Wolfram Language. (2010). JordanModelDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JordanModelDecomposition.html

BibTeX

@misc{reference.wolfram_2026_jordanmodeldecomposition, author="Wolfram Research", title="{JordanModelDecomposition}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/JordanModelDecomposition.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_jordanmodeldecomposition, organization={Wolfram Research}, title={JordanModelDecomposition}, year={2010}, url={https://reference.wolfram.com/language/ref/JordanModelDecomposition.html}, note=[Accessed: 13-June-2026]}