Use of geometric algebra: compound matrices and the determinant of the sum of two matrices

Prells, Uwe; Friswell, Michael I.; Garvey, Seamus D.

doi:10.1098/rspa.2002.1040

Cited by 15 publications

(13 citation statements)

References 14 publications

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“…Since det(∇ 2 h) is a 2 × 2 matrix, we can write it as [26], det(∇ 2 h) = det(M 1 ) + det(M 2 ) + tr(M † 1 M 2 ), where M † 1 is the adjugate of M 1 . From (38), it can easily be shown that det(M 1 ) ≥ 0.…”

Section: Discussionmentioning

confidence: 99%

Learning to Communicate in UAV-Aided Wireless Networks: Map-Based Approaches

Esrafilian

Gangula

Gesbert

2019

IEEE Internet Things J.

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We consider a scenario where an UAV-mounted flying base station is providing data communication services to a number of radio nodes spread over the ground. We focus on the problem of resource-constrained UAV trajectory design with (i) optimal channel parameters learning and (ii) optimal data throughput as key objectives, respectively. While the problem of throughput optimized trajectories has been addressed in prior works, the formulation of an optimized trajectory to efficiently discover the propagation parameters has not yet been addressed. When it comes to the communication phase, the advantage of this work comes from the exploitation of a 3D city map. Unfortunately, the communication trajectory design based on the raw map data leads to an intractable optimization problem. To solve this issue, we introduce a map compression method that allows us to tackle the problem with standard optimization tools. The trajectory optimization is then combined with a node scheduling algorithm. The advantages of the learning-optimized trajectory and of the map compression method are illustrated in the context of intelligent IoT data harvesting.

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Section: Discussionmentioning

confidence: 99%

Learning to Communicate in UAV-Aided Wireless Networks: Map-Based Approaches

Esrafilian

Gangula

Gesbert

2019

IEEE Internet Things J.

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“…We recall that the adjugate of an ðn À 1Þ Â ðn À 1Þ matrix is related to its ðn À 2Þth compound matrix and that the compound matrix of a diagonal matrix is also diagonal [22][23][24]. We also recall the Binet-Cauchy Theorem [22] which states that the compound matrix of a product is equal to the product of the compound matrices.…”

Section: A Placement Theorem For Defective Minorsmentioning

confidence: 99%

“…with the matrices V i :¼ C nÀ2 ðZ > S i ÞJ; W i :¼ C nÀ2 ðQS k ÞJ 2 C n 2 ÂðnÀ1Þ and where the vector ' j 2 C n 2 is the diagonal of the diagonal matrix C nÀ2 ðU j Þ: The matrix J is defined [22][23][24] as J :¼ ES where E is the rotated identity matrix,…”

Section: A Placement Theorem For Defective Minorsmentioning

confidence: 99%

On Pole–zero Placement by Unit-Rank Modification

Prells

Mottershead

Friswell

2003

Mechanical Systems and Signal Processing

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“…Furthermore, in [20] the above scalar product is expressed in terms of the compounds of A and B, i.e. both matrices are separated in the sense that the resulting expression consists of a sum of traces of products of their compound matrices.…”

Section: Determinant Of the Sum Of Two Matricesmentioning

confidence: 99%

“…For further information on existing literature and research topics about compound matrices, the reader can consult [4,[19][20][21][22] and the references therein.…”

Section: Computation Of An Uncorrupted Basementioning

confidence: 99%

Compound matrices: properties, numerical issues and analytical computations

Kravvaritis¹,

Mitrouli²

2008

Numer Algor

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This paper studies the possibility to calculate efficiently compounds of real matrices which have a special form or structure. The usefulness of such an effort lies in the fact that the computation of compound matrices, which is generally noneffective due to its high complexity, is encountered in several applications. A new approach for computing the Singular Value Decompositions (SVD's) of the compounds of a matrix is proposed by establishing the equality (up to a permutation) between the compounds of the SVD of a matrix and the SVD's of the compounds of the matrix. The superiority of the new idea over the standard method is demonstrated. Similar approaches with some limitations can be adopted for other matrix factorizations, too. Furthermore, formulas for the n − 1 compounds of Hadamard matrices are derived, which dodge the strenuous computations of the respective numerous large determinants. Finally, a combinatorial counting technique for finding the compounds of diagonal matrices is illustrated.

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Use of geometric algebra: compound matrices and the determinant of the sum of two matrices

Cited by 15 publications

References 14 publications

Learning to Communicate in UAV-Aided Wireless Networks: Map-Based Approaches

Learning to Communicate in UAV-Aided Wireless Networks: Map-Based Approaches

On Pole–zero Placement by Unit-Rank Modification

Compound matrices: properties, numerical issues and analytical computations

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