orthogonal

Examples of orthogonal in the following topics:

• A Geometrical Picture

  • Therefore all the elements in the null space are orthogonal to all the elements in the row space.
  • In mathematical terminology, the null space and the row space are orthogonal complements of one another.
  • Or, to say the same thing, they are orthogonal subspaces of $\mathbf{R}^{m}$ .
  • Similarly, vectors in the left null space of a matrix are orthogonal to all the columns of this matrix.
  • This means that the left null space of a matrix is the orthogonal complement of the column $\mathbf{R}^{n}$ .

• Superposition and orthogonal projection

  • But the solution is especially simple if the $\mathbf{x}_i$ are orthogonal.
  • In this case we can find the coefficients easily by projecting onto the orthogonal directions:
  • If the basis functions $q_i(x)$ are "orthogonal", then we should be able to compute the Fourier coefficients by simply projecting the function $f(x)$ onto each of the orthogonal "vectors" $q_i(x)$ .
  • Then we will say that two functions are orthogonal if their inner product is zero.
  • Now we simply need to show that the sines and cosines (or complex exponentials) are orthogonal.

• Some Special Matrices

  • A matrix $Q \in \mathbf{R}^{{n \times n}}$ is said to be orthogonal if $Q^TQ = I_n$ .
  • So why are these matrices called orthogonal?
  • An orthogonal matrix has an especially nice geometrical interpretation.
  • Therefore an orthogonal matrix maps a vector into another vector of the same norm.

• The Linear Algebra of the DFT

  • The matrix $Q$ is almost orthogonal.
  • We have said that a matrix $A$ is orthogonal if $A A^T = A^T A = I$, where $I$ is the N-dimensional identity matrix.
  • For complex matrices we need to generalize this definition slightly; for complex matrices we will say that $A$ is orthogonal if $(A^T)^* A = A (A^T)^* = I$ .
  • Once again, orthogonality saves us from having to solve a linear system of equations: since $Q^* = Q^{-1}$ , we have

• Experimental Design

  • Orthogonality: Orthogonality concerns the forms of comparison (contrasts) that can be legitimately and efficiently carried out.
  • Contrasts can be represented by vectors and sets of orthogonal contrasts are uncorrelated and independently distributed if the data are normal.
  • Because of this independence, each orthogonal treatment provides different information to the others.
  • If there are $T$ treatments and $T-1$ orthogonal contrasts, all the information that can be captured from the experiment is obtainable from the set of contrasts.
  • Outline the methodology for designing experiments in terms of comparison, randomization, replication, blocking, orthogonality, and factorial experiments

• Properties of Spherical Harmonics and Legendre Polynomials

  • The Legendre polynomials and the spherical harmonics satisfy the following "orthogonality" relations.
  • We will see shortly that these properties are the analogs for functions of the usual orthogonality relations you already know for vectors.
  • Notice that the second relation is slightly different than the others; it says that for any given value of $m$, the polynomials $P_{\ell m}$ and $P_{\ell ' m}$ are orthogonal.
  • To compute the coefficients of this expansion we use the orthogonality relation exactly as you would with an ordinary vector.

• Orthogonal decomposition of rectangular matrices

  • However, there is an amazingly useful generalization that pertains if we allow a different orthogonal matrix on each side of $A$ .
  • And since $S$ is symmetric it has orthogonal eigenvectors $\mathbf{w}_i$$\lambda _ i$ with real eigenvalues $\lambda _ i$
  • Keep in mind that the matrices $U$ and $V$ whose columns are the model and data eigenvectors are square (respectively $n \times n$$m \times m$ and $m \times m$ ) and orthogonal.

• Eigenvectors and Orthogonal Projections

  • Above we said that the matrices $V$ and $U$ were orthogonal so that $V^T V = V V^T = I_m$ and $U^T U = U U^T = I_n$ .

• A Matrix Appears

  • are orthogonal.
  • This orthogonality is an absolutely fundamental property of the natural modes of vibration of linear mechanical systems.

• Introduction to Least Squares

  • Since $A \mathbf{x_{ls}}$ is, by definition, confined to the column space of $A$ then $A\mathbf{x_{ls}} - \mathbf{y}$ (the error in fitting the data) must be in the orthogonal complement of the column space.
  • The orthogonal complement of the column space is the left null space, so $A\mathbf{x_{ls}} - \mathbf{y}$ must get mapped into zero by $A^T$ :
  • Before when we did orthogonal projections, the projecting vectors/matrices were orthogonal, so $A^TA$ term would have been the identity, but the outer product structure in $A\mathbf{x_{ls}}$ is evident.