Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e.,

where a_nn denotes the entry on the n-th row and n-th column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis. This characterization can be used to define the trace of a linear operator in general. Note that the trace is only defined for a square matrix (i.e., n × n).

The trace is related to the derivative of the determinant (see Jacobi's formula).

The term trace is a calque from the German Spur (cognate with the English spoor), which, as a function in mathematics, is often abbreviated to "tr".

Example

Let A be a matrix, with

Then

Properties

Basic properties

The trace is a linear mapping. That is,

for all square matrices A and B, and all scalars c.

A matrix and its transpose have the same trace:

This follows immediately from the fact that transposing a square matrix does not affect elements along the main diagonal.

Linear algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.

The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.

Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.

Total order

In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is transitive, antisymmetric, and total (this relation is denoted here by infix ≤). A set paired with a total order is called a totally ordered set, a linearly ordered set, a simply ordered set, or a chain.

If X is totally ordered under ≤, then the following statements hold for all a, b and c in X:

Antisymmetry eliminates uncertain cases when both a precedes b and b precedes a. A relation having the property of "totality" means that any pair of elements in the set of the relation are comparable under the relation. This also means that the set can be diagrammed as a line of elements, giving it the name linear.Totality also implies reflexivity, i.e., a ≤ a. Therefore, a total order is also a partial order. The partial order has a weaker form of the third condition. (It requires only reflexivity, not totality.) An extension of a given partial order to a total order is called a linear extension of that partial order.

Linear (album)

Linear is the name of the debut studio album by the pop/freestyle group Linear. It was released on March 21, 1990, by Atlantic Records. The album's first single, Sending All My Love, reached #5 on the Billboard Hot 100 making the album itself reach #52 on the Billboard 200. It also won a Gold certificate the same year, for selling more than five hundred thousand copies in the United States. The second single, "Don't You Come Cryin'", did not repeat the success of the previous single - though it achieved some prominence, reaching the 70th spot on the Billboard Hot 100 list. The album's last single, "Something Going On", failed to succeed.

Track listing

Charts

Singles - Billboard (North America)

References

Diatomic molecule

Diatomic molecules are molecules composed of only two atoms, of either the same or different chemical elements. The prefix di- is of Greek origin, meaning "two". If a diatomic molecule consists of two atoms of the same element, such as hydrogen (H₂) or oxygen (O₂), then it is said to be homonuclear. Otherwise, if a diatomic molecule consists of two different atoms, such as carbon monoxide (CO) or nitric oxide (NO), the molecule is said to be heteronuclear.

The only chemical elements that form stable homonuclear diatomic molecules at standard temperature and pressure (STP) (or typical laboratory conditions of 1 bar and 25 °C) are the gases hydrogen (H₂), nitrogen (N₂), oxygen (O₂), fluorine (F₂), and chlorine (Cl₂).

The noble gases (helium, neon, argon, krypton, xenon, and radon) are also gases at STP, but they are monatomic. The homonuclear diatomic gases and noble gases together are called "elemental gases" or "molecular gases", to distinguish them from other gases that are chemical compounds.

Trace

Trace may refer to:

Mathematics

Computer science and electronics

Trace (album)

Trace is the first album by Son Volt, released in 1995. The band was formed the previous year by Jay Farrar after the breakup of the influential alt-country band Uncle Tupelo. The album reached #166 on the Billboard 200 album chart and received extremely favorable reviews. According to Allmusic, "Throughout Son Volt's debut, Trace, the group reworks classic honky tonk and rock & roll, adding a desperate, determined edge to their performances. Even when they rock out, there is a palpable sense of melancholy to Farrar's voice, which lends a poignancy to the music." The album was in the top 10 of Rolling Stone's 1995 critics' list.

"Drown" was a minor college and rock radio hit. It charted at #10 on the Billboard Mainstream Rock Tracks chart and #25 on the Billboard Modern Rock Tracks chart. It remains their only single to chart on either of the charts.

Track listing

All of the songs were written by Farrar except "Mystifies Me", written by Ronnie Wood.