In an N-dimensional Hilbert space, can be written as an N × 1 column vector, and then A is an N × N matrix with complex entries.

The common FAT file system can have multiple files whose names differ only in case if the user uses a disk editor to edit the file names in the directory entries.

It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well.

A geometric interpretation can be given to the value of the determinant of a square matrix with real entries: the absolute value of the determinant gives the scale factor by which area or volume is multiplied under the associated linear transformation, while its sign indicates whether the transformation preserves orientation.

Although most often used for matrices whose entries are real or complex numbers, the definition of the determinant only involves addition, subtraction and multiplication, and so it can be defined for square matrices with entries taken from any commutative ring.

Equivalently, the determinant can be expressed as a sum of products of entries of the matrix where each product has n terms and the coefficient of each product is − 1 or 1 or 0 according to a given rule: it is a polynomial expression of the matrix entries.

The determinant of is denoted as, or it can be denoted directly in terms of the matrix entries by writing enclosing bars instead of brackets:

If the matrix entries are real numbers, the matrix can be used to represent two linear mappings: one that maps the standard basis vectors to the rows of, and one that maps them to the columns of.

POM contest demo entries can still be found on the Internet.

In it, the female narrator can be found wielding a pen and scribbling her diary entries under the most dramatic and unlikely of circumstances.

Note that due to the reduced number of colors in the image, there are obvious display issues. GIF is palette-based: the colors used in an image ( a frame ) in the file have their RGB values defined in a palette table that can hold up to 256 entries, and the data for the image refer to the colors by their indexes ( 0 – 255 ) in the palette table.

However, the intermediate entries can grow exponentially large, so it has exponential bit complexity.

The same technique can be used to map two-letter country codes like " us " or " za " to country names ( 26 < sup > 2 </ sup >= 676 table entries ), 5-digit zip codes like 13083 to city names ( 100000 entries ), etc.

In this example, if we can see that, the entries in the third row.

Mizuki Miyashita and Laura Moll note, " Copyrights on dictionaries are unusual because the entries in the dictionary are not copyrightable as the words themselves are facts, and facts can not be copyrighted.

Lighthouses mark dangerous coastlines, hazardous shoals, reefs, safe entries to harbors, and can also assist in aerial navigation.

This can be ( search just the named entry, typically used to read one entry ), ( entries immediately below the base DN ), or ( the entire subtree starting at the base DN ).

Server administrators can add additional schema entries in addition to the provided schema elements.

This requires a naming structure for LDAP entries so one can find a server holding a given DN or distinguished name, a concept defined in the X. 500 Directory and also used in LDAP.

In mathematics, with 2-or 3-dimensional vectors with real-valued entries, the idea of the " length " of a vector is intuitive and can easily be extended to any real vector space R < sup > n </ sup >.

Additional entries must be made from time to time.

This approach requires that: ( 1 ) each text word be separated into smaller elements to establish a correspondence between the occurrence and dictionary entries, and ( 2 ) the information retrieved from several entries in the dictionary be synthesized into a description of the particular word.

For example the entries in the Sales Journal are taken and a debit entry is made in each customer's account ( showing that the customer now owes us money ) and a credit entry might be made in the account for " Sale of class 2 widgets " ( showing that this activity has generated revenue for us ).

: The nominee must be at least past the legal driving age and free of mental defect ( Northcutt considers injury or death caused by mental defect to be tragic, rather than amusing, and routinely disqualifies such entries ).

Depending upon the initial entries by the user, the illusion of a human writer could be instantly dispelled, or could continue through several interchanges.

The various entries may or may not be separated by episodes.

In these cases, the uniformity criterion should hold for almost all typical subsets of entries that may be found in the table, not just for the global set of all possible entries.

The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows.

More generally, the general linear group of degree n over any field F ( such as the complex numbers ), or a ring R ( such as the ring of integers ), is the set of n × n invertible matrices with entries from F ( or R ), again with matrix multiplication as the group operation.

Usually the entries are numbers or expressions, but can even be matrices themselves ( see block matrix ).

If the entries of the matrix are real or complex numbers ( or from any other commutative ring ), then all four quantities are equal.

Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries.

The Napoleonic wars saw a reduction in the numbers of students and entries in the records for the purchase of muskets and other items for college members serving in the university corps.

Suppose M is an m × n matrix whose entries come from the field K, which is either the field of real numbers or the field of complex numbers.

In the special but common case in which M is just an m × m square matrix with positive determinant whose entries are plain real numbers, then U, V *, and Σ are m × m matrices of real numbers as well, Σ can be regarded as a scaling matrix, and U and V * can be viewed as rotation matrices.

The scale was made a standard for ship's log entries on Royal Navy vessels in the late 1830s and was adapted to non-naval use from the 1850s, with scale numbers corresponding to cup anemometer rotations.

In mathematics, the dot product, or scalar product ( or sometimes inner product in the context of Euclidean space ), is an algebraic operation that takes two equal-length sequences of numbers ( usually coordinate vectors ) and returns a single number obtained by multiplying corresponding entries and then summing those products.

While this provides a valid proof ( for matrices over the complex numbers ), the argument is not very satisfactory, since the identities represented by the theorem do not in any way depend on the nature of the matrix ( diagonalizable or not ), nor on the kind of entries allowed ( for matrices with real entries the diagonizable ones do not form a dense set, and it seems strange one would have to consider complex matrices to see that the Cayley – Hamilton theorem holds for them ).

Since B is also a matrix with polynomials in t as entries, one can for each i collect the coefficients of in each entry to form a matrix B < sub > i </ sub > of numbers, such that one has

By the finite-dimensional spectral theorem, such operators can be associated with an orthonormal basis of the underlying space in which the operator is represented as a diagonal matrix with entries in the real numbers.

Solo numbers by members of an established group were not eligible for the award as " separate entries from the duo or group performances.

The pseudoinverse is defined and unique for all matrices whose entries are real or complex numbers.

As a group of matrices, H < sub > n </ sub > ( or H < sub > n </ sub >( R ) to indicate this is the Heisenberg group over the ring R or real numbers ) is defined as the group of square matrices of size n + 2 with entries in R:

However, this approach also has a disadvantage: because entries are arranged in alphabetical order, so are the code numbers.

For the 2006 to 2007 school year, the NFL authorized a trial system to award extra entries into the national tournament to those tournaments with exceptionally large numbers of, or exceptional growth in, members and degrees.