scholarly journals The evaluation of eigenvalues and eigenvectors of real symmetric matrices by simultaneous iteration

1970 ◽  
Vol 13 (1) ◽  
pp. 76-80 ◽  
Author(s):  
M. Clint
Keyword(s):  
Symmetric Matrices ◽  
Eigenvalues And Eigenvectors ◽  
Simultaneous Iteration

scholarly journals Tolerance and Nature of Residual Refraction in Symmetric Power Space as Principal Lens Powers and Meridians Change

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Herven Abelman ◽  
Shirley Abelman
Keyword(s):  
Symmetric Power ◽  
Symmetric Matrices ◽  
Eigenvalues And Eigenvectors ◽  
Real Scalar ◽  
Power Change ◽  
Refractive Correction ◽  
Dioptric Power ◽  
The One ◽  
Residual Power

Unacceptable principal powers in well-centred lenses may require a toric over-refraction which differs in nature from the one where correct powers have misplaced meridians. This paper calculates residual (over) refractions and their natures. The magnitude of the power of the over-refraction serves as a general, reliable, real scalar criterion for acceptance or tolerance of lenses whose surface relative curvatures change or whose meridians are rotated and cause powers to differ. Principal powers and meridians of lenses are analogous to eigenvalues and eigenvectors of symmetric matrices, which facilitates the calculation of powers and their residuals. Geometric paths in symmetric power space link intended refractive correction and these carefully chosen, undue refractive corrections. Principal meridians alone vary along an arc of a circle centred at the origin and corresponding powers vary autonomously along select diameters of that circle in symmetric power space. Depending on the path of the power change, residual lenses different from their prescription in principal powers and meridians are pure cross-cylindrical or spherocylindrical in nature. The location of residual power in symmetric dioptric power space and its optical cross-representation characterize the lens that must be added to the compensation to attain the power in the prescription.

scholarly journals Eigenvalues and Eigenvectors for 3×3 Symmetric Matrices: An Analytical Approach

2020 ◽  
pp. 106-118
Author(s):  
Abu Bakar Siddique ◽  
Tariq A. Khraishi
Keyword(s):  
Analytical Solutions ◽  
Analytical Approach ◽  
Numerical Solutions ◽  
Linear Equations ◽  
Symmetric Matrices ◽  
Matrix Equations ◽  
Eigenvalues And Eigenvectors ◽  
Engineering Problems ◽  
The Matrix ◽  
Research Problems

Research problems are often modeled using sets of linear equations and presented as matrix equations. Eigenvalues and eigenvectors of those coupling matrices provide vital information about the dynamics/flow of the problems and so needs to be calculated accurately. Analytical solutions are advantageous over numerical solutions because numerical solutions are approximate in nature, whereas analytical solutions are exact. In many engineering problems, the dimension of the problem matrix is 3 and the matrix is symmetric. In this paper, the theory behind finding eigenvalues and eigenvectors for order 3×3 symmetric matrices is presented. This is followed by the development of analytical solutions for the eigenvalues and eigenvectors, depending on patterns of the sparsity of the matrix. The developed solutions are tested against some examples with numerical solutions.

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