You are here:

Question

A square matrix A is called orthogonal if (A^T)A = I_n.
(a) Show that

A =          [cos(theta), -sin(theta)]
[sin(theta), cos(theta)]

is orthogonal.

b) Assume that A, B are orthogonal matrices of the same size. Show that AB is also orthogonal.
(c) Let vector v_1, vector v_2,...., vector v_n be the columns of an orthogonal matrix A. Show that the vector v_i's are mutually perpendicular and unit vectors.

a) do the matrix multiplication and use the trig identity

cos^2(theta) + sin^2(theta) = 1.

b) AB orthogonal means (AB)^T(AB) = I. Using properties of the transpose

(AB)^T(AB) = B^TA^T(AB) = B^T(A^TA)B = B^TB = I.

c) let the matrix A be written (aij) where 1 ≤ i,j ≤ n are the indices of the matrix entries. Also, let v_i = (i1, i2 ... in)^T be the ith column vector of A. Matrix multiplication can be written

A^TA = (aij)^T(aij) = ∑∑vi･vj, sum over i and j

i.e., equals the dot-product of the ith row and jth column. Since the entries of A^TA are 1 along the diagonal (i=j) and zero otherwise (i≠j)

∑∑vi･vj = δij = Kronecker delta function ⇒ vi are mutually perpendicular and unit length.

Volunteer

randy patton

Expertise

college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

Experience

26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

Publications
J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

Education/Credentials
M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

Past/Present Clients
Also an Expert in Oceanography