#AtoZChallenge2015: Orthogonal

Orthogonal projectionThe word orthogonal conjures up memories of wonderful maths classes, many years ago, as well as more obscure readings, much later, of absconse topological subjects. Geometry was one of the great pleasures of my youth: yes, we are all different!

I quote from: Barile, Margherita. “Orthogonal.” From MathWorld–A Wolfram Web Resource, created by Eric W. Weissteinhttp://mathworld.wolfram.com/Orthogonal.html

“In elementary geometry, orthogonal is the same as perpendicular. Two lines or curves are orthogonal if they are perpendicular at their point of intersection. Two vectors v and w of the real plane R^2 or the real space R^3 are orthogonal iff their dot product v·w=0. This condition has been exploited to define orthogonality in the more abstract context of the n-dimensional real space R^n.

More generally, two elements v and w of an inner product space E are called orthogonal if the inner product of v and w is 0. Two subspaces V and W of E are called orthogonal if every element of V is orthogonal to every element of W. The same definitions can be applied to any symmetric or differential k-form and to any Hermitian form.”

For those of my readers so inclined, Wikipedia has an interesting article on matrices!

“In linear algebra, an orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e.

Q^\mathrm{T} Q = Q Q^\mathrm{T} = I,

where I is the identity matrix.

This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse:

Q^\mathrm{T}=Q^{-1}, \,

An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q*) and therefore normal (Q*Q = QQ*) in the reals. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the dot product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation or reflection. In other words, it is a unitary transformation.

The set of n × n orthogonal matrices forms a group O(n), known as the orthogonal group. The subgroup SO(n) consisting of orthogonal matrices with determinant +1 is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation.

The complex analogue of an orthogonal matrix is a unitary matrix.”

Image: orthogonal projection at http://english.rejbrand.se/algosim/visualisation.asp?id=orthogonal_projection

Daily Prompt: The Transporter

Tell us about a sensation — a taste, a smell, a piece of music — that transports you back to childhood.

Staircase I bought this hand-wash at the supermarket, it appealed to me: tea-tree oil and coal tar fragrance…  Coal tar…  As I washed my hands it came back to me, through a mist of memories, as in a dream…

The narrow streets, the ancient doorways with stones for riders to dismount, and metal rings on the walls to secure the horses, the steep wooden stairways that appear to rise for ever to mysterious lofts…  And then I am there again, a boy still, in the small medieval town, the cobbled lanes, and you, in the cold air of an early eastern spring, and the smell of coal tar the town uses to repair the side walks – so long ago…

Nostalgia overcomes me and I start writing