minor

Examples of minor in the following topics:

• Cofactors, Minors, and Further Determinants

  • The cofactor of an entry $(i,j)$ of a matrix $A$ is the signed minor of that matrix.
  • To know what the signed minor is, we need to know what the minor of a matrix is.
  • Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors.
  • The determinant of any matrix can be found using its signed minors.
  • Explain how to use minor and cofactor matrices to calculate determinants

• Parts of an Ellipse

  • The minor axis of the ellipse is the shortest width across it.
  • The minor axis has length $2b$.
  • Its endpoints are the minor axis vertices, with coordinates $(h, k \pm b)$.
  • This diagram of a horizontal ellipse shows the ellipse itself in red, the center $C$ at the origin, the focal points at $\left(+f,0\right)$ and $\left(-f,0\right)$, the major axis vertices at $\left(+a,0\right)$ and $\left(-a,0\right)$, the minor axis vertices at $\left(0,+b\right)$ and $\left(0,-b\right)$.

• Ellipses

  • The distance between antipodal points on the ellipse, or pairs of points whose midpoint is at the center of the ellipse, is maximum along the major axis or transverse diameter, and a minimum along the perpendicular minor axis or conjugate diameter.
  • The semi-major axis (denoted by a in the figure) and the semi-minor axis (denoted by b in the figure) are one half of the major and minor axes, respectively.
  • These are sometimes called (especially in technical fields) the major and minor semi-axes, or major radius and minor radius.
  • The area enclosed by an ellipse is $\pi ab$, where a and b are one-half of the ellipse's major and minor axes respectively.

• Parts of a Hyperbola

  • The co-vertices correspond to $b$, the "minor semi-axis length", and have coordinates $(h,k+b)$ and $(h,k-b)$.
  • The major and minor axes $a$ and $b$, as the vertices and co-vertices, describe a rectangle that shares the same center as the hyperbola, and has dimensions $2a \times 2b$.
  • Both its major and minor axis values are equal, so that $a = b = \sqrt{2m}$.

• Standard Equations of Hyperbolas

  • A conjugate axis of length 2b, corresponding to the minor axis of an ellipse, is sometimes drawn on the non-transverse principal axis; its endpoints ±b lie on the minor axis at the height of the asymptotes over/under the hyperbola's vertices.

• What is a Matrix?

  • The term "matrix" (Latin for "womb", derived from mater—mother) was coined by James Joseph Sylvester in $1850$, who understood a matrix as an object giving rise to a number of determinants today called minors, that is to say, determinants of smaller matrices that are derived from the original one by removing columns and rows.  

• Types of Conic Sections

  • A minor axis, which is the shortest width across the ellipse
  • where $(h,k)$ are the coordinates of the center, $2a$ is the length of the major axis, and $2b$ is the length of the minor axis.

• Applications of Circles and Ellipses

  • The major axis length is $2a = 3$, and the minor axis length is $\displaystyle{2b = \frac{2}{5}}$.

• What Are Conic Sections?

  • The transverse axis is also called the major axis, and the conjugate axis is also called the minor axis.