If A=[ 0 -tan(α)/(2) tan(α)/(2) 0 ] and I is the identity matrix of order 2, show

If A=[ 0 -tan(α)/(2) tan(α)/(2) 0 ] and I is...

Key Concepts

Tangent Half-Angle Substitution

The tangent half-angle substitution is a trigonometric technique that expresses sine and cosine in terms of tan(?/2). This identity is particularly useful when relating a rotation matrix to a skew?symmetric matrix via the Cayley transform, as it simplifies the algebraic expressions and links the parameters of the matrix representation to the rotation angle.

Rotation Matrix

A rotation matrix in two dimensions rotates vectors through a specified angle and is typically expressed as [[cos ?, -sin ?], [sin ?, cos ?]]. It is an orthogonal matrix with determinant one and encapsulates the idea of rotating a point in the plane. Rotation matrices serve as the backbone for many applications in physics, engineering, and computer graphics.

Skew-Symmetric Matrices

A skew?symmetric matrix is one for which the transpose equals the negative of the matrix. In the 2×2 case, such matrices have the form [[0, -a], [a, 0]]. These matrices arise naturally in the study of rotations and have properties that make them useful in representing infinitesimal rotations, serving as generators of rotation groups.

Cayley Transform

The Cayley transform is a mapping that transforms a skew?symmetric matrix into an orthogonal matrix (typically a rotation matrix) without eigenvalue ?1. It is defined in this context by expressions like (I + A)(I ? A)?¹ or its variants. This transform provides a bridge between the algebra of matrices and the geometry of rotations, enabling one to express rotations in a rational form.

Transcript

00:01 Problem number 18 we have problem number 18 question says we have given matrix a 0 minus 10 alpha by 2 then 10 alpha by 2 and 0 okay and i is an identical matrix of order 2 so basically i is given 1 0 0 this is order 2 that 2 cross 2 matrix and we have to show some thing we have to show that i plus a equal to i minus a and cos alpha minus sine alpha sine alpha and causal so this is basically a calculative problem calculation type problem and very easy so left -hand side is i plus a so i nt matrix plus a so if we add them up so 0 plus 1 will be zero simply one zero person will be one sorry so simply this will be added up minus ten alpha by two and then ten alpha by two and one this is simple left and side and right inside we have to calculate i minus alpha we have to calculate first so i minus alpha so one minus 0 is 0 and then 0 minus minus 10 alpha by 2 is simply 10 alpha by 2 and then 0 minus 10 alpha by 2 is minus 1 0 is 1 so this is our i minus alpha now you have to multiply this with the given matrix which is cos alpha minus x alpha and alpha sign alpha and cost alpha now let us start doing so if we have to multiply we'll multiply like this so 0 into cos alpha is 0 plus cos alpha into 10 alpha cause alpha 10 alpha by 2 similarly first 1st 2 and second column 0 plus 10 alpha by 2 sorry now this was 0 into cos alpha 0 alpha and it should be sine alpha sign alpha and here it should be 10 alpha by 2 and cause alpha again this should be cause alpha into minus 10 alpha by 2 and plus sine alpha and secondly minus minus plus so simple sign alpha and 10 alpha by 2 plus cos alpha so this is the product now we have to make it in the term of this okay there is a mistake here that it should have been one here it is one so it should have been one one so it could have been okay cause alpha if multiply with cause alpha and here minus sin alpha so it should be cos alpha k and then here it should be minus sine alpha okay now comes the calculation part okay now we can write it as cos alpha plus sine alpha into 10 alpha by 2 could be written as sine alpha by 2 by cos alpha by 2…

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