) This orthogonality condition can be used to derive the formula for the IDFT from the definition of the DFT, and is equivalent to the unitarity property below.

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.

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.

Another orthogonality relation is the closure equation:

For the spherical Bessel functions the orthogonality relation is:

( 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 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.

However, there is a certain amount of design philosophy similarity ( e. g., considerable orthogonality and flexible addressing modes ), some assembly language syntax resemblance, as well as opcode mnemonic similarity, but the 6809 is a derivative of the 6800 whereas the 68000 was a totally new design.

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 second is an elegant argument using orthogonality and is based upon: Mackiw, G. ( 1995 ).

* Logical independence is geometric orthogonality ;

Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to vectors.

Hence orthogonality of vectors is an extension of the concept of perpendicular vectors into higher-dimensional spaces.

The converse is not true ; having a determinant of + 1 is no guarantee of orthogonality, even with orthogonal columns, as shown by the following counterexample.

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.

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 ).

In general, uncorrelatedness is not the same as orthogonality, except in the special case where either X or Y has an expected value of 0.

The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces.

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

The extension to either Euclidean or Minkowskian geometry is achieved by adding various further axioms of orthogonality, etc

In particular, we now have the concepts of orthogonal complement and orthogonality of subrepresentations.

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.

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:

which again from the orthogonality of leads to the following equation

which orthogonality demands satisfy the three equations

At the time, computer design focused on adding as many instructions as possible to the machine's CPU, a concept known as " orthogonality ", which made programs smaller and more efficient in use of memory.

which, by orthogonality of the eigenvectors, becomes:

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

Darboux trihedron, consisting of a point P, and a triple of orthogonality | orthogonal unit vector s e < sub > 1 </ sub >, e < sub > 2 </ sub >, and e < sub > 3 </ sub > which is adapted to a surface in the sense that P lies on the surface, and e < sub > 3 </ sub > is perpendicular to the surface.

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

Nevertheless they retain their mutual orthogonality which is what facilitates the decomposition.

Subtracting the two, we get the Galerkin orthogonality relation for the error, which is the error between the solution of the original problem,, and the solution of the Galerkin equation,

It also made the computers themselves fantastically complex, and in an era when many CPUs were hand-wired from individual transistors, the cost of additional orthogonality was often very high.