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Properties of matrices

1 Properties of matrices In this section, we review some basic concepts of matrix theory and some fundamental properties of matrices, focusing on those that will be needed in later sections. Matrices and vectors A matrix is a rectangular
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Solving systems of linear equations

Solving systems of linear equations Solving a set of simultaneous linear equations is a fundamental problem that occurs in diverse applications. A linear system can be expressed as a matrix equation in which each matrix or vector element bel
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Symmetric positive-definite matrices and least-squares approximation

Symmetric positive-definite matrices and least-squares approximation Symmetric positive-definite matrices have many interesting and desirable properties. For example, they are nonsingular, and LU decomposition can be performed on them without o
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Computational geometry

COMPUTATIONAL GEOMETRY Computational geometry is the branch of computer science that studies algorithms for solving geometric problems. In modern engineering and mathematics, computational geometry has applications in, among other fields, c
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Inverting matrices

Inverting matrices Although in practice we do not generally use matrix inverses to solve systems of linear equations, preferring instead to use more numerically stable techniques such as LUP decomposition, it is sometimes necessary to compu
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Polynomials and the fft

POLYNOMIALS AND THE FFT The straightforward method of adding two polynomials of degree n takes (n) time, but the straightforward method of multiplying them takes (n2) time. In this chapter, we shall show how the Fast Fourier Transform, or FFT,
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Strassen's algorithm for matrix multiplication

Strassen's algorithm for matrix multiplication This section presents Strassen's remarkable recursive algorithm for multiplying n n matrices that runs in (nlg 7) = O(n2.81) time. For sufficiently large n, therefore, it outperforms the naive
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Number-theoretic algorithms

NUMBER-THEORETIC ALGORITHMS Number theory was once viewed as a beautiful but largely useless subject in pure mathematics. Today number-theoretic algorithms are used widely, due in part to the invention of cryptographic schemes based on large pr
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Linear algebra theorems

LINEAR ALGEBRA THEOREMS 1.1              A homogeneous system of m equations in n variables either has a unique solution (zero solution) or has infinitely many solutions, de
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Conics - a class of plain curves

C O N I C S Conics reprezent a class of plain curves with special properites, which have numerous applications in various domains. The mathematicians of the elenic era obtained non-degenerated conics by the intersection of a rotati
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Trigonometry - Maths Extension

Trigonometry       Trigonometric Ratios       Exact Values & Triangles       Trigonometric Identities       ASTC Rule  &
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Geometrical Characteristics of the Beam Cross-Section

Geometrical Characteristics of the Beam Cross-Section The definitions of the beam and cross-section were specified in the previous lectures. Some geometrical characteristics of the cross-section, such as the area and moments of iner
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Complex Numbers


Complex Numbers The beauty of physics and mathematics is that everything is somehow related to everything else, and that there are so many ways of looking at the same thing, each providing new insights. Take for instance the Fourier transform.
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Mathematical Model of the OFDM Signal

Mathematical Model of the OFDM Signal In order to perform the mathematical analysis of OFDM signals, the most convenient strategy is to use the signal theory related to the orthogonal expansion of signa
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A DIFFERENT DEMONSTRATION OF THE PYTHAGOREAN THEOREM USING THE POWER OF THE POINT WITH RESPECT TO A CIRCLE

A DIFFERENT DEMONSTRATION OF THE PYTHAGOREAN THEOREM USING THE POWER OF THE POINT WITH RESPECT TO A CIRCLE In the present paper I will give / offer you not only a different demonstration of the Pythagorean Theorem but also
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Maths and Vibrations Tutorial Sheet 4

Maths and Vibrations Tutorial Sheet 4 1) Find the limit of the following functions as x tends to zero, i) ii) iii) 2) Differentiate the following equations with respe
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Maths and Vibrations Tutorial Sheet 2

Maths and Vibrations Tutorial Sheet 2 1) Sketch the functions sin(x), cos(x) and tan(x) and also cosec(x), sec(x) and cot(x) between -p and p. Note the angles where they cross the axes and indicate any asymptotes. Which are odd or even f
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Maths and Vibrations: Tutorial Sheet 9 Solutions

Maths and Vibrations: Tutorial Sheet 9 Solutions 1) b) Inertia 2) b) (see course book part 2) 3) 4) 5) 6) Total ene
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Tutorial Sheet 7 Solutions

Tutorial Sheet 7 Solutions With most transforms simply refer to a table (e.g. page 57,58). 1) i) ii) iii) iv)
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Irish Maths Test

Irish Maths Test   An Irishman wants a job, but the foreman won't hire him until he passes a little math test. Here is your first question, the foreman said.  'Without using numbers, represent the number 9.' 'W
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