Information about Diagonal Matrix
In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero. Thus, the matrix D = (di,j) with
n columns and n rows is diagonal if:
For example, the following matrix is diagonal:
The term diagonal matrix may sometimes refer to a rectangular diagonal matrix, which is an m-by-n matrix with only the entries of the form di,i possibly non-zero; for example,
However, in the remainder of this article we will consider only square matrices.
Any diagonal matrix is also a symmetric matrix. Also, if the entries come from the field R or C, then it is a normal matrix as well.
Equivalently, we can define a diagonal matrix as a matrix that is both upper- and lower-triangular.
The identity matrix In and any square zero matrix are diagonal. A one-dimensional matrix is always diagonal.
A diagonal matrix with all its main diagonal entries equal is a scalar matrix, that is, a scalar multiple λI of the identity matrix I. Its effect on a vector is scalar multiplication by λ. The scalar matrices are the center of the algebra of matrices: that is, they are precisely the matrices that commute with all other square matrices of the same size.
and for matrix multiplication,
The diagonal matrix diag(a1,...,an) is invertible if and only if the entries a1,...,an are all non-zero. In this case, we have
In particular, the diagonal matrices form a subring of the ring of all n-by-n matrices.
Multiplying the matrix A from the left with diag(a1,...,an) amounts to multiplying the i-th row of A by ai for all i; multiplying the matrix A from the right with diag(a1,...,an) amounts to multiplying the i-th column of A by ai for all i.
The adjugate of a diagonal matrix is again diagonal.
A square matrix is diagonal if and only if it is triangular and normal.
In fact, a given n-by-n matrix A is similar to a diagonal matrix (meaning that there is a matrix X such that XAX-1 is diagonal) if and only if it has n linearly independent eigenvectors. Such matrices are said to be diagonalizable.
Over the field of real or complex numbers, more is true. The spectral theorem says that every normal matrix is unitarily similar to a diagonal matrix (if AA* = A*A then there exists a unitary matrix U such that UAU* is diagonal). Furthermore, the singular value decomposition implies that for any matrix A, there exist unitary matrices U and V such that UAV* is diagonal with positive entries.
For example, the following matrix is diagonal:
The term diagonal matrix may sometimes refer to a rectangular diagonal matrix, which is an m-by-n matrix with only the entries of the form di,i possibly non-zero; for example,
, or 
.
However, in the remainder of this article we will consider only square matrices.
Any diagonal matrix is also a symmetric matrix. Also, if the entries come from the field R or C, then it is a normal matrix as well.
Equivalently, we can define a diagonal matrix as a matrix that is both upper- and lower-triangular.
The identity matrix In and any square zero matrix are diagonal. A one-dimensional matrix is always diagonal.
A diagonal matrix with all its main diagonal entries equal is a scalar matrix, that is, a scalar multiple λI of the identity matrix I. Its effect on a vector is scalar multiplication by λ. The scalar matrices are the center of the algebra of matrices: that is, they are precisely the matrices that commute with all other square matrices of the same size.
Matrix operations
The operations of matrix addition and matrix multiplication are especially simple for diagonal matrices. Write diag(a1,...,an) for a diagonal matrix whose diagonal entries starting in the upper left corner are a1,...,an. Then, for addition, we have- diag(a1,...,an) + diag(b1,...,bn) = diag(a1+b1,...,an+bn)
and for matrix multiplication,
- diag(a1,...,an) · diag(b1,...,bn) = diag(a1b1,...,anbn).
The diagonal matrix diag(a1,...,an) is invertible if and only if the entries a1,...,an are all non-zero. In this case, we have
- diag(a1,...,an)-1 = diag(a1-1,...,an-1).
In particular, the diagonal matrices form a subring of the ring of all n-by-n matrices.
Multiplying the matrix A from the left with diag(a1,...,an) amounts to multiplying the i-th row of A by ai for all i; multiplying the matrix A from the right with diag(a1,...,an) amounts to multiplying the i-th column of A by ai for all i.
Other properties
The eigenvalues of diag(a1, ..., an) are a1, ..., an. The unit vectors e1, ..., en form a basis of eigenvectors. The determinant of diag(a1, ..., an) is the product a1...an.The adjugate of a diagonal matrix is again diagonal.
A square matrix is diagonal if and only if it is triangular and normal.
Uses
Diagonal matrices occur in many areas of linear algebra. Because of the simple description of the matrix operation and eigenvalues/eigenvectors given above, it is always desirable to represent a given matrix or linear map by a diagonal matrix.In fact, a given n-by-n matrix A is similar to a diagonal matrix (meaning that there is a matrix X such that XAX-1 is diagonal) if and only if it has n linearly independent eigenvectors. Such matrices are said to be diagonalizable.
Over the field of real or complex numbers, more is true. The spectral theorem says that every normal matrix is unitarily similar to a diagonal matrix (if AA* = A*A then there exists a unitary matrix U such that UAU* is diagonal). Furthermore, the singular value decomposition implies that for any matrix A, there exist unitary matrices U and V such that UAV* is diagonal with positive entries.
See also
External links
References
- Roger A. Horn and Charles R. Johnson, Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-30586-1 (hardback), ISBN 0-521-38632-2 (paperback).
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations.
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In linear algebra, the main diagonal (sometimes leading diagonal) of a square matrix is the diagonal which runs from the top left corner to the bottom right corner. For example, the following matrix has 1s down its main diagonal:
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In linear algebra, a symmetric matrix is a square matrix, A, that is equal to its transpose
The entries of a symmetric matrix are symmetric with respect to the main diagonal (top left to bottom right).
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The entries of a symmetric matrix are symmetric with respect to the main diagonal (top left to bottom right).
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field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
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normal matrix if
where A* is the conjugate transpose of A. (If A is a real matrix, A*=AT and so it is normal if ATA = AAT.
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- A*A=AA*
where A* is the conjugate transpose of A. (If A is a real matrix, A*=AT and so it is normal if ATA = AAT.
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In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. Because matrix equations with triangular matrices are easy to solve they are very important in numerical analysis.
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In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. Because matrix equations with triangular matrices are easy to solve they are very important in numerical analysis.
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In linear algebra, the identity matrix or unit matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere.
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In mathematics, particularly linear algebra, a zero matrix is a matrix with all its entries being zero. Some examples of zero matrices are
The set of m×n matrices with entries in a ring K forms a ring .
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The set of m×n matrices with entries in a ring K forms a ring .
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In linear algebra, the identity matrix or unit matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere.
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In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). Note that scalar multiplication is different than scalar product
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The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. More specifically:
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- The center of a group G consists of all those elements x in G
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This article gives an overview of the various ways to perform matrix multiplication.
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Ordinary matrix product
By far the most important way to multiply matrices is the usual matrix multiplication...... Click the link for more information.
This article gives an overview of the various ways to perform matrix multiplication.
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Ordinary matrix product
By far the most important way to multiply matrices is the usual matrix multiplication...... Click the link for more information.
invertible or non-singular if there exists an n-by-n matrix such that
where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.
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where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.
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In mathematics, a subring is a subset of a ring, which contains the multiplicative identity and is itself a ring under the same binary operations. Naturally, those authors who do not require rings to contain a multiplicative identity do not require subrings to possess the identity
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eigenvector of the transformation and the blue vector is not. Since the red vector was neither stretched nor compressed, its eigenvalue is 1. All vectors with the same vertical direction - i.e., parallel to this vector - are also eigenvectors, with the same eigenvalue.
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In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1 (the unit length). A unit vector is often written with a superscribed caret or “hat”, like this (pronounced "i-hat").
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basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. In other words, a basis is a linearly independent spanning set.
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eigenvector of the transformation and the blue vector is not. Since the red vector was neither stretched nor compressed, its eigenvalue is 1. All vectors with the same vertical direction - i.e., parallel to this vector - are also eigenvectors, with the same eigenvalue.
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In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A
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In linear algebra, the adjugate or classical adjoint of a square matrix is a matrix which plays a role similar to the inverse of a matrix; it can however be defined for any square matrix without the need to perform any divisions.
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In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
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similar if there exists an invertible n-by-n matrix P over K such that
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- P −1AP = B.
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In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent.
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In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e. if there exists an invertible matrix P such that P −1AP is a diagonal matrix.
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field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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In mathematics, a complex number is a number of the form
where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized (that
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