Appendix AMatrix Forms and Relationships

The following matrix forms, properties, and relationships are useful in the derivation of state estimators [18,52]. Vectors are depicted with small letters as a, x, … and matrices with capital letters as X, Y, … The matrix forms and relationships are united in groups depending on properties and applications.

A.1 Derivatives

The following derivatives of matrix and vectors products and traces of the products allow deriving state estimators in a shorter way.

(A.1)StartFraction partial-differential x Superscript upper T Baseline a Over partial-differential x EndFraction equals StartFraction partial-differential a Superscript upper T Baseline x Over partial-differential x EndFraction equals a comma
(A.2)StartFraction partial-differential x Superscript upper T Baseline upper B x Over partial-differential x EndFraction equals left-parenthesis upper B plus upper B Superscript upper T Baseline right-parenthesis x comma
(A.3)StartFraction partial-differential Over partial-differential upper X EndFraction trace left-parenthesis upper X upper A right-parenthesis equals upper A Superscript upper T Baseline comma
(A.4)StartFraction partial-differential Over partial-differential upper X EndFraction trace left-parenthesis upper X Superscript upper T Baseline upper A right-parenthesis equals upper A comma
(A.5)StartFraction partial-differential Over partial-differential upper X EndFraction trace left-parenthesis upper X Superscript upper T Baseline upper B upper X right-parenthesis equals upper B upper X plus upper B Superscript upper T Baseline upper X comma
(A.6)StartFraction partial-differential Over partial-differential upper X EndFraction trace left-parenthesis upper X upper B upper X Superscript upper T Baseline right-parenthesis equals upper X upper B Superscript upper T Baseline plus upper X upper B period

A.2 Matrix Identities

There are several matrix identities that are useful in the representation of the inverse of the sum of matrices.

The Woodbury identities [59]:

(A.7)left-parenthesis upper A plus upper C upper B upper C Superscript upper T Baseline right-parenthesis Superscript negative 1 Baseline equals upper A Superscript negative 1 Baseline minus upper A Superscript negative 1 Baseline upper C left-parenthesis upper B Superscript negative 1 Baseline plus upper C Superscript upper T Baseline upper A Superscript negative 1 Baseline upper C right-parenthesis Superscript negative 1 Baseline upper C Superscript upper T Baseline upper A Superscript negative 1 Baseline comma
(A.8)

For positive definite matrices and , there is

(A.9)

The Kailath ...

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