augmented matrix

Examples of augmented matrix in the following topics:

• Simplifying Matrices With Row Operations

  • Augmented matrix: an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices.
  • A triangular matrix is one that is either lower triangular or upper triangular.
  • A matrix that is both upper and lower triangular is a diagonal matrix.
  • Use elementary row operations on the augmented matrix $[A|b]$ to transform $A$ to upper triangle form.
  • Use elementary row operations to put a matrix in simplified form

• Matrices and Row Operations

  • Row multiplication (scale): Multiply a row of a matrix by a nonzero constant.
  • Row addition (pivot): Add to one row of a matrix some multiple of another row.
  • Since the matrix is essentially the coefficients and constants of a linear system, the three row operations preserve the matrix.
  • Because these operations are reversible, the augmented matrix produced always represents a linear system that is equivalent to the original.
  • There are several specific algorithms to row-reduce an augmented matrix, the simplest of which are Gaussian elimination and Gauss-Jordan elimination.

• Further Simplification of Matrices

  • It is a variation of Gaussian elimination, which places zeros below each pivot in the matrix, starting with the top row and working downwards.
  • Both Gauss–Jordan and Gaussian elimination have time complexity of order $O({ n }^{ 3 })$ for an n by n full rank matrix (using Big O Notation), but the order of magnitude of the number of arithmetic operations (there are roughly the same number of additions and multiplications/divisions) used in solving an n by n matrix by Gauss-Jordan elimination is ${n}^{3}$, whereas that for Gaussian elimination is $\frac{2n}{3}$.
  • The steps of Gauss-Jordan elimination are very similar to that of Gaussian elimination, the main difference being that we will work in diagonal form instead of putting the augmented matrix into upper triangle form.
  • Use elementary row operations on matrix [A|b] to transform A into diagonal form.
  • A matrix is in reduced row echelon form (also called row canonical form) if it is the result of a Gauss–Jordan elimination.

• Breast Augmentation and Reduction

  • Breast augmentation is surgical enhancement of the breasts to increase size, while reduction is surgical removal of tissue to decrease size.
  • Breast augmentation denotes the breast implant and fat-graft mammoplasty procedures for correcting the defects, and for enhancing the size, form, and feel of a woman's breasts.
  • The surgical implantation approach effects global breast augmentation using either a saline-filled or a silicone-filled prosthetic breast; and it might also consist of corrections effected with transplanted skin flaps.
  • The fat-transfer approach effects the augmentation, and corrects the contour defects of the breast hemisphere with grafts of autologous adipocyte fat tissue.
  • In non-implant breast augmentation practice, some fat-graft injection approaches feature tissue engineering, which is the pre-operative external tissue expansion of the recipient site.

• Augmented and Diminished Chords

  • Because they don't contain a perfect fifth, augmented and diminished chords have an unsettled feeling and are normally used sparingly.
  • An augmented chord is built from two major thirds, which adds up to an augmented fifth.
  • Listen closely to an augmented triad (http://cnx.org/content/m10890/latest/choaug.mp3) and a diminished triad (http://cnx.org/content/m10890/latest/chodim.mp3).
  • For example, if, in an augmented G sharp major chord, you rewrite the D double sharp as an E natural, the triad becomes an E augmented chord.

• The Period

  • Augmented-sixth chords are so named because of the augmented sixth that occurs between le and fi, the two scale-degrees that are present in every augmented-sixth chord.
  • The Italian augmented-sixth chord is the simplest augmented-sixth chord, with only three members: le, do, and fi.
  • The French augmented-sixth chord has four members: le, do, re, andfi.
  • The German augmented-sixth chord has four members: le, do, me, and fi.
  • The Swiss augmented-sixth chord has four members: le, do, ri, and fi.

• Chromatically Altered Subdominant Chords

  • Augmented-sixth chords are so named because of the augmented sixth that occurs between le and fi, two scale-degrees that are present in every augmented-sixth chord.
  • The Italian augmented-sixth chord is the simplest augmented-sixth chord, with only three members: le, do, and fi.
  • The French augmented-sixth chord has four members: le, do, re, and fi.
  • The German augmented-sixth chord has four members: le, do, me, and fi.
  • The Swiss augmented-sixth chord has four members: le, do, ri, and fi.

• Summary of the Types of Intervals

  • An augmented interval is one half step larger than the perfect or major interval.
  • Inversions of augmented intervals are diminished, and inversions of diminished intervals are augmented.

• Muscle Development

  • When the FGF runs out, the myoblasts cease division and secrete fibronectin onto their extracellular matrix.
  • Satellite cells are able to differentiate and fuse to augment existing muscle fibers and to form new ones.
  • They are able to differentiate and fuse to augment existing muscle fibers and to form new ones.

• What is a Matrix?

  • The matrix has a long history of application in solving linear equations.
  • A matrix with m rows and n columns is called an m × n matrix or $m$-by-$n$ matrix, while m and n are called its dimensions.
  • A matrix which has the same number of rows and columns is called a square matrix.
  • In some contexts, such as computer algebra programs, it is useful to consider a matrix with no rows or no columns, called an empty matrix.
  • Each element of a matrix is often denoted by a variable with two subscripts.