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Data Compression in Health Monitoring
Published in Rajarshi Gupta, Dwaipayan Biswas, Health Monitoring Systems, 2019
Sourav Kumar Mukhopadhyay, Rajarshi Gupta
where A is the matrix containing the MBioSigs, UTU = I, VTV = I, and the columns of Ur × r = [u1, u2,…ur] and Vc × c = [v1, v2,…vr] matrices are the left and right singular vectors, respectively. The matrix S is diagonal with real positive entries, which are called the singular values (σ). Singular values are the square roots of eigenvalues from either U or V matrix. The total information-energy of the A matrix can be mathematically expressed as
Dealing with Generalized PDEs
Published in Guigen Zhang, Introduction to Integrative Engineering, 2017
Before we discuss solving PDEs with time-dependent terms, let us first review the two relevant concepts, eigenvalues and eigenvectors. Eigenvalues are a special set of scalars that are associated with a matrix equation, and they are sometimes known as characteristic values. Similarly, eigenvectors are a special set of vectors associated with a matrix equation, and they are sometimes referred to as characteristic vectors. In daily life, the determination of the eigenvalues and eigenvectors of a physical or engineering system is extremely important in situations such as stability analysis, rotating bodies, and free vibration. Each eigenvalue is associated with a corresponding eigenvector.
Dimension Reduction Breaking the Curse of Dimensionality
Published in Chong Ho Alex Yu, Data Mining and Exploration, 2022
In the context of statistical analysis, vectors help us to understand the relationships among variables. An eigenvalue has a numeric property, while an eigenvector has a directional property. These properties together define the characteristics of a variable or a component. Eigenvalues and eigenvectors are mathematical objects in which inputs are largely unaffected by a mathematical transformation. Again, take vector-based graphics as an example. Whenever the image is rescaled, the algorithm recreates the vectors, and thus even if the image is enlarged 2,000%, it remains crystal sharp.
Damping of Inter-Area Oscillations Using TCPS Based Delay Compensated Robust WADC
Published in Electric Power Components and Systems, 2023
Abhineet Prakash, Kundan Kumar, S. K. Parida
The symbols A, B, and C are used to refer to the system, input, and output matrices, respectively. The state, input, and output of the system are represented by the variables x, u, and y, respectively. A diagonal matrix can be created where the values on the diagonal represent the eigenvalues of the system matrix A. The left and right eigenvectors of A as and respectively, such that: where I denote the identity matrix of order n. Let z be the modal transformation of x defined by x = Mz. Using Eqs. (14) and (15), the following expression can be derived:
New gradient methods with adaptive stepsizes by approximate models
Published in Optimization, 2023
Zexian Liu, Hongwei Liu, Ting Wang
According to (1), we know , which implies that there exist at least mutually orthogonal vectors such that As a result, which implies that at least eigenvalues of all equal to . Now we calculate the other two eigenvalues and . According to the fact that the trace of a matrix equals to the sum of all eigenvalues of the matrix, by some simple algebraic operations we have Using the fact that the determinant of a matrix equals the product of all eigenvalues of the matrix, we can easily obtain Combining (13) and (14), after some simple algebraic operations we obtain that which completes the proof.
PSO Based reduced order modelling of autonomous AC microgrid considering state perturbation
Published in Automatika, 2020
Mudita Juneja, S. K. Nagar, Soumya R. Mohanty
A real eigenvalue corresponds to a non-oscillatory mode whereas, a complex eigenvalue is an oscillatory mode. The frequency of oscillation for oscillatory modes is obtained from the imaginary part of eigenvalues. The real part of eigenvalues gives the damping effect of that root on the system [22]. All the eigenvalues of the system in Table 3, have negative real parts. Thus, the necessary condition for small signal stability is fulfilled, i.e. . The evaluation of the participation factor of a system is crucial to further enhance the stability. The participation matrix combines the right eigenvectors and left eigenvectors to measure the degree of association between all the state variables of the system and the system modes. The eigenvalue sensitivity is found out from this matrix, which gives the contribution or influence of a certain state variable to a certain mode [23].