# JordanModelDecomposition

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:
•  , continuous-time system ,. discrete-time system
• The new state matrix 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:

## Scope(4)

The Jordan decomposition of a second-order system:

The Jordan decomposition of a discrete-time system:

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

Repeated poles appear in Jordan blocks:

## 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:

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:

## Properties & Relations(3)

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

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

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

## Possible Issues(1)

JordanModelDecomposition does not support descriptor systems:

Use KroneckerModelDecomposition to separate the modes of the system:

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_2023_jordanmodeldecomposition, author="Wolfram Research", title="{JordanModelDecomposition}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/JordanModelDecomposition.html}", note=[Accessed: 18-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_jordanmodeldecomposition, organization={Wolfram Research}, title={JordanModelDecomposition}, year={2010}, url={https://reference.wolfram.com/language/ref/JordanModelDecomposition.html}, note=[Accessed: 18-April-2024 ]}