In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, matrix calculus is a specialized notation for doing multivariable calculus

Multivariable calculus

Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable: the differentiated and integrated functions involve multiple variables, rather than just one....

, especially over spaces of matrices

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

, where it defines the matrix derivative. This notation was to describe systems of differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s, and taking derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

s of matrix-valued functions with respect to matrix variables. This notation is commonly used in statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

 and engineering

Engineering

Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

, while the tensor index notation is preferred in physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

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Note this article uses an alternate definition for vector and matrix calculus than the form often encountered within the field of estimation theory

Estimation theory

Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the...

 and pattern recognition

Pattern recognition

In machine learning, pattern recognition is the assignment of some sort of output value to a given input value , according to some specific algorithm. An example of pattern recognition is classification, which attempts to assign each input value to one of a given set of classes...

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