In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is , while an example of a 3×3 diagonal matrix is. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix.
A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size). Its determinant is the product of its diagonal values.
As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix D = (di,j) with n columns and n rows is diagonal if
However, the main diagonal entries are unrestricted.
The term diagonal matrix may sometimes refer to a , which is an m-by-n matrix with all the entries not of the form di,i being zero. For example:
or
More often, however, diagonal matrix refers to square matrices, which can be specified explicitly as a . A square diagonal matrix is a symmetric matrix, so this can also be called a .
The following matrix is square diagonal matrix:
If the entries are real numbers or complex numbers, then it is a normal matrix as well.
In the remainder of this article we will consider only square diagonal matrices, and refer to them simply as "diagonal matrices".
A diagonal matrix can be constructed from a vector using the operator:
This may be written more compactly as .
The same operator is also used to represent block diagonal matrices as where each argument is a matrix.
The operator may be written as:
where represents the Hadamard product and is a constant vector with elements 1.
The inverse matrix-to-vector operator is sometimes denoted by the identically named where the argument is now a matrix and the result is a vector of its diagonal entries.
The following property holds:
A diagonal matrix with equal diagonal entries is a scalar matrix; that is, a scalar multiple λ of the identity matrix I.
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In linear algebra, the identity matrix of size is the square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1. The identity matrix is often denoted by , or simply by if the size is immaterial or can be trivially determined by the context.
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is , while an example of a 3×3 diagonal matrix is. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix. A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size).
Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.
Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.
Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.
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