( Although mutual orthogonality is the only condition, these vectors are usually constructed for ease of decoding, for example columns or rows from Walsh matrices.

The orthogonality of the DFT is now expressed as an orthonormality condition ( which arises in many areas of mathematics as described in root of unity ):

Suppose the entries of Q are differentiable functions of t, and that t = 0 gives Q = I. Differentiating the orthogonality condition

This property provides a simple method to test the condition of orthogonality.

and satisfy the orthogonality condition

They satisfy the orthogonality condition

He demonstrated his principle using vector space geometry based on an “ orthogonality condition ” which only worked in systems where the velocities were defined as a single vector or tensor, and thus, as he wrote at p. 347, was “ impossible to test by means of macroscopic mechanical models ”, and was, as he pointed out, invalid in “ compound systems where several elementary processes take place simultaneously ”.

Because this equation holds for all vectors, p, we conclude that every rotation matrix, Q, satisfies the orthogonality condition,

While analogous to ordinary orthogonality, particularly as it is known with ordinary complex number arithmetic, this condition is more subtle.

Assuming 0 ≤ m ≤ &# 8467 ;, they satisfy the orthogonality condition for fixed m:

Also, they satisfy the orthogonality condition for fixed ℓ:

This orthogonality relation can then be used to extract the coefficients in the Fourier – Bessel series, where a function is expanded in the basis of the functions J < sub > α </ sub >( x u < sub > α, m </ sub >) for fixed α and varying m.

* Every Jacobi-like polynomial sequence can have its domain shifted and / or scaled so that its interval of orthogonality is, and has Q

* Every Laguerre-like polynomial sequence can have its domain shifted, scaled, and / or reflected so that its interval of orthogonality is, and has Q =

* Every Hermite-like polynomial sequence can have its domain shifted and / or scaled so that its interval of orthogonality is, and has Q

It can be shown that orthogonality of functions is a generalization of the concept of orthogonality of vectors.

The Gauss sum can thus be written as a linear combination of Gaussian periods ( with coefficients χ ( a )); the converse is also true, as a consequence of the orthogonality relations for the group ( Z / nZ )< sup >×</ sup >.

This orthogonality can best be understood in a thought experiment: Consider a model of a population of animals such as crocodiles or tangle web spiders.

So, the direct product of two irreducible representations can also be turned into a unitary representations and now, we have the neat orthogonality property allowing us to decompose the direct product into a direct sum of irreducible representations ( we're also using the property that for finite dimensional representations, if you keep taking proper subrepresentations, you'll hit an irreducible representation eventually.

The relation of hyperbolic orthogonality actually applies to classes of parallel lines in the plane, where any particular line can represent the class.

which can be readily verified using the orthogonality relationship described below.

These theorems can be proven using the orthogonality property.

A shorter, non-numerical example can be found in orthogonality principle.

where the last step can be seen to follow e. g. from a Taylor series expansion, and due to the orthogonality of the states and we have

The resultant throughput reduction can be partly compensated with a large tone set so that each symbol represents several data bits ; a long symbol interval allows these tones to be packed more closely in frequency while maintaining orthogonality.

The image above shows the sign to be negative: to maintain orthogonality, if du < sub > ρ </ sub > is positive with dθ, then du < sub > θ </ sub > must decrease.

These systems differ in their orthogonality, the Geek Code is very orthogonal in the computer science sense ( may be projected onto basis vectors ), where the Yerkes system is very orthogonal in the taxonomic sense ( represent mutually exclusive classes ).

the interval of orthogonality is, still have two parameters to be determined.

Substituting these expressions into the entropy integral, exchanging the order of integration and summation, and using the orthogonality of the cosines, the entropy may be written:

Conversely, if the Doppler spread is large while the delay spread is small, then a shorter symbol period may permit coherent tone detection and the tones must be spaced more widely to maintain orthogonality.

However, in practice ( as the calculations are performed in floating point arithmetic where inaccuracy is inevitable ), the orthogonality is quickly lost and in some cases the new vector could even be linearly dependent on the set that is already constructed.

There were no specific input or output instructions ; the PDP-11 used memory-mapped I / O and so the same move instruction was used ; orthogonality even enabled moving data directly from an input device to an output device.

The " up tack " symbol ( U + 22A5: ⊥) used by philosophers and logicians ( see contradiction ) also appears, but is often avoided due to its usage for orthogonality.

The lemma is named after Issai Schur who used it to prove Schur orthogonality relations and develop the basics of the representation theory of finite groups.

Reciprocity is also a basic lemma that is used to prove other theorems about electromagnetic systems, such as the symmetry of the impedance matrix and scattering matrix, symmetries of Green's functions for use in boundary-element and transfer-matrix computational methods, as well as orthogonality properties of harmonic modes in waveguide systems ( as an alternative to proving those properties directly from the symmetries of the eigen-operators ).

Because Bessel's equation becomes Hermitian ( self-adjoint ) if it is divided by x, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions.

where · denotes the dot product, see also orthogonality for more information.

Another general goal was to provide every possible addressing mode for every instruction, known as orthogonality, to ease compiler implementation.

Here orthogonality is important not only for reducing A < sup > T </ sup > A = ( R < sup > T </ sup > Q < sup > T </ sup >) QR to R < sup > T </ sup > R, but also for allowing solution without magnifying numerical problems.

Using the orthogonality relation where is the Kronecker delta, we simplify the above three terms for each to see

Affine geometry provides the basis for Euclidean structure when perpendicular lines are defined, or the basis for Minkowski geometry through the notion of hyperbolic orthogonality.

For α = β = 0, these are called the Legendre polynomials ( for which the interval of orthogonality is and the weight function is simply 1 ):

If the polynomials f are such that the term on the left is zero, and for, then the orthogonality relationship will hold:

Using the orthogonality relations derived above, we find that the q characters for the q inequivalent irreducible representations forms a basis set.

See Lewis & Wilson or Felsagen for the radius-tangent orthogonality.

The VAX-11 extended the PDP-11's orthogonality to all data types, including floating point numbers ( although instructions such as ' ADD ' was divided into data-size dependent variants such as ADDB, ADDW, ADDL, ADDP, ADDF for add byte, word, longword, packed BCD and single-precision floating point, respectively ).

This implies the desired orthogonality relationship for the characters: i. e.,

A different version of the orthogonality principle exists for linear MMSE estimators.