Category Archives: what-is

What Is a Matrix Square Root?

A square root of an matrix is any matrix such that . For a scalar (), there are two square roots (which are equal if ), and they are real if and only if is real and nonnegative. For , … Continue reading

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What Is IEEE Standard Arithmetic?

The IEEE Standard 754, published in 1985 and revised in 2008 and 2019, is a standard for binary and decimal floating-point arithmetic. The standard for decimal arithmetic (IEEE Standard 854) was separate when it was first published in 1987, but … Continue reading

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What Is Floating-Point Arithmetic?

A floating-point number system is a finite subset of the real line comprising numbers of the form where is the base, is the precision, and is the exponent. The system is completely defined by the four integers , , , … Continue reading

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What Is Rounding?

Rounding is the transformation of a number expressed in a particular base to a number with fewer digits. For example, in base 10 we might round the number to , which can be described as rounding to three significant digits … Continue reading

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What Is a Random Orthogonal Matrix?

Various explicit parametrized formulas are available for constructing orthogonal matrices. To construct a random orthogonal matrix we can take such a formula and assign random values to the parameters. For example, a Householder matrix is orthogonal and symmetric and we … Continue reading

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What Is a Correlation Matrix?

In linear algebra terms, a correlation matrix is a symmetric positive semidefinite matrix with unit diagonal. In other words, it is a symmetric matrix with ones on the diagonal whose eigenvalues are all nonnegative. The term comes from statistics. If … Continue reading

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What Is a Hadamard Matrix?

A Hadamard matrix is an matrix with elements and mutually orthogonal columns. For example, is a Hadamard matrix. A necessary condition for an Hadamard matrix to exist with is that is divisible by , but it is not known if … Continue reading

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What Is an Orthogonal Matrix?

A real, square matrix is orthogonal if (the identity matrix). Equivalently, . The columns of an orthogonal matrix are orthonormal, that is, they have 2-norm (Euclidean length) and are mutually orthogonal. The same is true of the rows. Important examples … Continue reading

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What Is a Generalized Inverse?

The matrix inverse is defined only for square nonsingular matrices. A generalized inverse is an extension of the concept of inverse that applies to square singular matrices and rectangular matrices. There are many definitions of generalized inverses, all of which … Continue reading

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What Is a Matrix?

A matrix is a rectangular array of numbers on which certain algebraic operations are defined. Matrices provide a convenient way of encapsulating many numbers in a single object and manipulating those numbers in useful ways. An matrix has rows and … Continue reading

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