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Study of matrices and their algebraic properties
In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular
Matrix_analysis
Array of numbers
analysis. Square matrices, matrices with the same number of rows and columns, play a major role in matrix theory. The determinant of a square matrix is
Matrix_(mathematics)
Ray tracing technique
Ray transfer matrix analysis (also known as ABCD matrix analysis) is a mathematical form for performing ray tracing calculations in sufficiently simple
Ray_transfer_matrix_analysis
Method of data analysis
data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis. Factor analysis typically
Principal_component_analysis
Boston Consulting Group business analysis method
growth–share matrix (also known as the product portfolio matrix, Boston Box, BCG-matrix, Boston matrix, Boston Consulting Group portfolio analysis and portfolio
Growth–share_matrix
Matrix with shifting rows
In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to
Toeplitz_matrix
Matrix equal to its conjugate-transpose
mathematics, more precisely in linear algebra, a Hermitian matrix (or self-adjoint matrix) is a square matrix that is equal to its own conjugate transpose—that
Hermitian_matrix
Dimension of the column space of a matrix
In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal number
Rank_(linear_algebra)
Norm on a vector space of matrices
such norms are referred to as matrix norms. Matrix norms behave in certain ways like the distance from the zero matrix. They are distinguished from the
Matrix_norm
Complex matrix whose conjugate transpose equals its inverse
In linear algebra, an invertible complex square matrix U is unitary if its matrix inverse U−1 equals its conjugate transpose U*, that is, if U ∗ U = U
Unitary_matrix
Components of a chemical sample other than the substance of interest
In chemical analysis, matrix refers to the components of a sample other than the analyte of interest. The matrix can have a considerable effect on the
Matrix_(chemical_analysis)
Measure of covariance of components of a random vector
covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the
Covariance_matrix
Matrix equal to its transpose
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, A is symmetric ⟺ A = A T . {\displaystyle A{\text{
Symmetric_matrix
Matrix in which most of the elements are zero
In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict
Sparse_matrix
Algorithms for matrix decomposition
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra
Non-negative matrix factorization
Non-negative_matrix_factorization
In matrix analysis, Kreiss matrix theorem relates the so-called Kreiss constant of a matrix with the power iterates of this matrix. It was originally
Kreiss_matrix_theorem
Matrix that commutes with its conjugate transpose
more amenable to analysis. The spectral theorem states that a matrix is normal if and only if it is unitarily similar to a diagonal matrix; implying in particular
Normal_matrix
Mathematical operation in linear algebra
columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number
Matrix_multiplication
Method for analysing qualitative data
approaches (these include approaches like framework analysis, template analysis and matrix analysis) and reflexive approaches. They first described their
Thematic_analysis
Matrix decomposition
(also known as eigenvalue decomposition or EVD) is a factorization of a matrix A {\displaystyle A} into a canonical form given by A = V D V T {\displaystyle
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Matrix decomposition
complex matrix into a rotation, followed by a scaling, followed by another rotation. It generalizes the eigendecomposition of a square normal matrix with
Singular_value_decomposition
Statistical method
Formal concept analysis Independent component analysis Non-negative matrix factorization Q methodology Recommendation system Root cause analysis Facet theory
Factor_analysis
Business planning and analysis technique
and strategic management, SWOT analysis (also known as the SWOT matrix, TOWS, WOTS, WOTS-UP, and situational analysis) is a decision-making technique
SWOT_analysis
Matrix of inner products of vectors
Gram matrix (or Gramian matrix, Gramian) of vectors v 1 , … , v n {\displaystyle v_{1},\dots ,v_{n}} in an inner product space is the Hermitian matrix of
Gram_matrix
Mathematics concept
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose
Hadamard_matrix
Matrix with nonzero elements on the main diagonal and the diagonals above and below it
In linear algebra, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal/lower diagonal (the first
Tridiagonal_matrix
Matrices similar to diagonal matrices
linear algebra, a square matrix A {\displaystyle A} is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists
Diagonalizable_matrix
For a square matrix, the transpose of the cofactor matrix
classical adjoint adj(A) of a square matrix A is the transpose of its cofactor matrix. It is occasionally known as adjunct matrix, or "adjoint", though that normally
Adjugate_matrix
Representation of a matrix as a product
particular class of problems. In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For example, when solving
Matrix_decomposition
American mathematician
(born January 19, 1942) is an American mathematician specializing in matrix analysis. He was research professor of mathematics at the University of Utah
Roger_Horn
Matrix defined using smaller matrices called blocks
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices
Block_matrix
Matrix which differs from the identity matrix by one elementary row operation
mathematics, an elementary matrix is a square matrix obtained from the application of a single elementary row operation to the identity matrix. The elementary matrices
Elementary_matrix
Matrix with no negative elements
In mathematics, a nonnegative matrix, written X ≥ 0 , {\displaystyle \mathbf {X} \geq 0,} is a matrix in which all the elements are equal to or greater
Nonnegative_matrix
Vectors mapped to 0 by a linear map
), Addison Wesley, ISBN 978-0-321-28713-7. Meyer, Carl D. (2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics
Kernel_(linear_algebra)
Vector spaces associated to a matrix
range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range
Row_and_column_spaces
Matrix with the same number of rows and columns
mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n {\displaystyle
Square_matrix
Table layout for visualizing performance; also called an error matrix
In machine learning, a confusion matrix, also known as error matrix, is a specific table layout that allows visualization of the performance of an algorithm
Confusion_matrix
Matrix of geometric progressions
In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row:
Vandermonde_matrix
Quantitative study of size and shape
-μετρία metria, "measurement") or morphometry refers to the quantitative analysis of form, a concept that encompasses size and shape. Morphometric analyses
Morphometrics
In mathematics, invariant of square matrices
square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the
Determinant
Special kind of square matrix
in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix L and an upper
Triangular_matrix
Academic journal
The SIAM Journal on Matrix Analysis and Applications is a peer-reviewed scientific journal covering matrix analysis and its applications. The relevant
SIAM Journal on Matrix Analysis and Applications
SIAM_Journal_on_Matrix_Analysis_and_Applications
Idempotent linear transformation from a vector space to itself
Least-squares spectral analysis Orthogonalization Properties of trace Meyer, pp 386+387 Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis, second edition
Projection_(linear_algebra)
Matrix of second derivatives
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function
Hessian_matrix
Branch of mathematics
Survey of Matrix Theory and Matrix Inequalities, Dover Publications, ISBN 978-0-486-67102-4 Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied
Linear_algebra
Concepts from linear algebra
applications, for example in stability analysis, vibration analysis, atomic orbitals, facial recognition, and matrix diagonalization. In essence, an eigenvector
Eigenvalues_and_eigenvectors
Matrix whose conjugate transpose is its negative (additive inverse)
Charles R. (1985), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6. Meyer, Carl D. (2000), Matrix Analysis and Applied Linear Algebra
Skew-Hermitian_matrix
Unsolved problem in matrix analysis
Crouzeix's conjecture is an unsolved problem in matrix analysis. It was proposed by Michel Crouzeix in 2004, and it can be stated as follows: ‖ f ( A
Crouzeix's_conjecture
Matrix operation generalizing exponentiation of scalar numbers
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems
Matrix_exponential
Stochastic matrix representing links between entities
A Google matrix is a particular stochastic matrix that is used by Google's PageRank algorithm. The matrix represents a graph with edges representing links
Google_matrix
Matrix whose only nonzero elements are on its main diagonal
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
Diagonal_matrix
Mathematical operation
square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix product BB is
Square_root_of_a_matrix
Property of a mathematical matrix
In mathematics, a symmetric matrix M {\displaystyle M} with real entries is positive-definite if the real number x T M x {\displaystyle \mathbf {x} ^{\mathsf
Definite_matrix
Matrix with a multiplicative inverse
algebra, an invertible matrix (non-singular, non-degenerate or regular) is a square matrix that has an inverse. In other words, if a matrix is invertible, it
Invertible_matrix
1999 film by the Wachowskis
The Matrix is a 1999 science fiction action film written and directed by the Wachowskis. The first installment in the Matrix film series, it stars Keanu
The_Matrix
Theorem of matrix ranks
algebra, the Woodbury matrix identity – named after Max A. Woodbury – says that the inverse of a rank-k correction of some matrix can be computed by doing
Woodbury_matrix_identity
Matrix with every entry equal to one
In mathematics, a matrix of ones or all-ones matrix is a matrix with every entry equal to one. For example: J 2 = [ 1 1 1 1 ] , J 3 = [ 1 1 1 1 1 1 1
Matrix_of_ones
Subclass of matrices
In mathematics, a square matrix is said to be diagonally dominant if, for every row of the matrix, the magnitude of the diagonal entry in a row is greater
Diagonally_dominant_matrix
Equivalence under a change of basis (linear algebra)
matrices A and B are called similar if there exists an invertible n-by-n matrix P such that B = P − 1 A P . {\displaystyle B=P^{-1}AP.} Two matrices are
Matrix_similarity
Technique used in brand marketing and product management
similar to BCG analysis, but more complex with nine cells rather than four. Like in BCG analysis, a two-dimensional portfolio matrix is created. However
GE_multifactorial_analysis
Sum of elements on the main diagonal
In linear algebra, the trace of a square matrix A, denoted tr(A), is defined as a sum of the elements on its main diagonal, a 11 + a 22 + ⋯ + a n n {\displaystyle
Trace_(linear_algebra)
Kind of square matrix in linear algebra
algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries
Hessenberg_matrix
Tool used for project and business process management
A responsibility assignment matrix, RACI matrix (responsible, accountable, consulted and informed matrix; /ˈreɪsi/) or linear responsibility chart, is
Responsibility assignment matrix
Responsibility_assignment_matrix
Technique in natural language processing
Principal Component Analysis (PCA) on the matrix A, except that PCA subtracts off the means. PCA loses the sparseness of the A matrix, which can make it
Latent_semantic_analysis
Matrix that, squared, equals itself
linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix A {\displaystyle A} is idempotent
Idempotent_matrix
American media franchise
The Matrix is an American cyberpunk media franchise consisting of four feature films, beginning with The Matrix (1999) and continuing with three sequels
The_Matrix_(franchise)
Tool in linear algebra and matrix analysis
linear algebra, the theory of matrices, numerical analysis, and statistics. It is defined for a block matrix. Suppose p, q are nonnegative integers such that
Schur_complement
In linear algebra, relation between 3 dimensions
a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity of M; and the dimension of
Rank–nullity_theorem
Matrix whose entries are all minors of another matrix
mathematics, a (multiplicative) compound matrix is a matrix whose entries are all minors, of a given size, of another matrix. Compound matrices are closely related
Compound_matrix
Risk assessment comparing the likelihood of a risk to its severity
A risk matrix is a matrix that is used during risk assessment to define the level of risk by considering the category of likelihood (often confused with
Risk_matrix
Numerical methods for matrix eigenvalue calculation
numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue
Eigenvalue_algorithm
Indian mathematician
include major contributions to matrix inequalities, matrices in graph theory, generalized inverses, and matrix analysis. In addition to the numerous research
Ravindra_Bapat
Decomposition in multilinear algebra
Ill-Posedness of the Best Low-Rank Approximation Problem". SIAM Journal on Matrix Analysis and Applications. 30 (3): 1084–1127. arXiv:math/0607647. doi:10.1137/06066518x
Tensor_rank_decomposition
Topics referred to by the same term
Ray transfer matrix analysis in geometric optics, a mathematical method for performing ray tracing calculations. Transfer-matrix method (combinatorics)
Transfer-matrix_method
Mathematical operation on invertible matrices
mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization
Logarithm_of_a_matrix
Elementwise product of two matrices
a matrix of the multiplied corresponding elements. This operation can be thought as a "naive matrix multiplication" and is different from the matrix product
Hadamard_product_(matrices)
Matrix used to study systems of ordinary differential equations
monodromy matrix is the fundamental matrix of a system of ODEs evaluated at the period of the coefficients of the system. It is used for the analysis of periodic
Monodromy_matrix
Polynomial whose roots are the eigenvalues of a matrix
algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It
Characteristic_polynomial
American mathematician (born 1948)
mathematics at College of William and Mary. The books Matrix Analysis and Topics in Matrix Analysis, co-written by him with Roger Horn, are standard texts
Charles_Royal_Johnson
Fictional character in The Matrix
Trinity is a fictional character in the Matrix franchise. She is portrayed by Carrie-Anne Moss in the films. In the gameplay segments of Path of Neo, she
Trinity_(The_Matrix)
Square matrix constructed from a monic polynomial
In linear algebra, the Frobenius companion matrix of the monic polynomial p ( x ) = c 0 + c 1 x + ⋯ + c n − 1 x n − 1 + x n {\displaystyle p(x)=c_{0}+c_{1}x+\cdots
Companion_matrix
Mathematical concept in algebra
(2012). Matrix Analysis. Cambridge University Press. p. 70. ISBN 9780521839402. Horn, Roger A.; Johnson, Charles R. (2012). Matrix Analysis. Cambridge
Commuting_matrices
Square roots of the eigenvalues of the self-adjoint operator
Horn and C. R. Johnson. Topics in Matrix Analysis. Cambridge University Press, Cambridge, 1991. Chap. 3 X. Zhan. Matrix Inequalities. Springer-Verlag, Berlin
Singular_value
Matrix factorisation in mathematics
Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily similar to an upper triangular matrix whose diagonal
Schur_decomposition
Algorithmic runtime requirements for matrix multiplication
algorithm for matrix multiplication? More unsolved problems in computer science In theoretical computer science, the computational complexity of matrix multiplication
Computational complexity of matrix multiplication
Computational_complexity_of_matrix_multiplication
Hungarian and American mathematician and physicist (1903–1957)
in Matrix Analysis. Cambridge University Press. p. 139. ISBN 0-521-30587-X. Horn & Johnson 2013, p. 335. Bhatia, Rajendra (1997). Matrix Analysis. Graduate
John_von_Neumann
Signal processing computational method
Mathematics portal Blind deconvolution Factor analysis Hilbert spectrum Image processing Non-negative matrix factorization (NMF) Nonlinear dimensionality
Independent component analysis
Independent_component_analysis
2003 film by the Wachowskis
The Matrix Reloaded is a 2003 American science fiction action film written and directed by the Wachowskis. It is the sequel to The Matrix (1999) and the
The_Matrix_Reloaded
bases. A matrix is consimilar to itself, its complex conjugate, its transpose and its adjoint matrix. Every matrix is consimilar to a real matrix and to
Matrix_consimilarity
Lens with a thickness that is negligible
with the paraxial approximation in techniques such as ray transfer matrix analysis. The focal length, f, of a lens in air is given by the lensmaker's
Thin_lens
Multilinear extension of principal component analysis
..., RN ) approximation of higher-order tensors". SIAM Journal on Matrix Analysis and Applications. 21 (4): 1324–1342. doi:10.1137/s0895479898346995
Multilinear principal component analysis
Multilinear_principal_component_analysis
Indian mathematician, author, and educator (born 1952)
include matrix inequalities, calculus of matrix functions, means of matrices, and connections between harmonic analysis, geometry and matrix analysis. He
Rajendra_Bhatia
Mathematical bound
Hadamard's inequality, see the proof of Theorem 7.8.5 in Horn and Johnson's Matrix Analysis. Assume that A and C are positive-definite. We have A − 1 {\displaystyle
Fischer's_inequality
Matrix of values of explanatory variables
particular in regression analysis, a design matrix, also known as model matrix or regressor matrix and often denoted by X, is a matrix of values of explanatory
Design_matrix
Specialized notation for multivariable calculus
the setting of functional analysis to take derivatives with respect to vectors. In the case that a matrix function of a matrix is Fréchet differentiable
Matrix_calculus
Measure of the "size" of linear operators
ISBN 978-0-07-054236-5. OCLC 21163277. Bhatia, Rajendra (1997). Matrix Analysis. Graduate Texts in Mathematics. Vol. 169. New York, NY: Springer New
Operator_norm
Small angle approximation in geometric optics
used in Gaussian optics and first-order ray tracing. Ray transfer matrix analysis is one method that uses the approximation. In some cases, the second-order
Paraxial_approximation
American mathematician (1928–2022)
including matrix analysis, complex analysis, approximation theory, and scientific computation. He was the author of the classic textbook Matrix Iterative
Richard_S._Varga
Matrix consisting of a single row or column
m\times 1} matrix consisting of a single column of m {\displaystyle m} entries. Similarly, a row vector is a 1 × n {\displaystyle 1\times n} matrix, consisting
Row_and_column_vectors
Theorem in linear algebra
functional analysis Z-matrix (mathematics) – Square matrix whose off-diagonal entries are nonpositive M-matrix – Matrix in mathematics P-matrix – Complex
Perron–Frobenius_theorem
MATRIX ANALYSIS
MATRIX ANALYSIS
Girl/Female
Maori
The Maori form of April.
Male
English
 English form of Roman Latin Martinus, MARTIN means "of/like Mars." Compare with another form of Martin.
Male
English
Anglicized form of Irish Gaelic MainchÃn, MANNIX means "little monk."
Male
English
Pet form of English Matthew, MATTIE means "gift of God." Compare with feminine Mattie.
Male
French
French and German form of Greek Mattathias, MATHIS means "gift of God."
Female
Finnish
Finnish form of Greek Margarites, MAARIT means "pearl."
Male
English
Pet form of English Martin, MARTIE means "of/like Mars."
Female
German
Pet form of German Katarine, KATRIN means "pure."
Female
Finnish
Pet form of Finnish Katariina, KATRI means "pure."
Female
English
English form of Latin Viatrix, BEATRIX means "voyager (through life)."
Female
Finnish
Finnish form of Greek Maria, MAARIA means "obstinacy, rebelliousness" or "their rebellion."Â
Female
English
Pet form of English Matilda, MATTIE means "mighty in battle." Compare with masculine Mattie.
Male
Italian
Italian form of Hebrew Mattithyah, MATTIA means "gift of God."
Girl/Female
Arabic, Australian, Basque, French, Latin
Lady; Feminine of Martin; Warlike
Male
French
 French form of Roman Latin Martinus, MARTIN means "of/like Mars." Compare with another form of Martin.
Male
Hungarian
Czech and Hungarian form of Greek Patrikios, PATRIK means "patrician, of noble descent."
Surname or Lastname
English (of Welsh origin)
English (of Welsh origin) : variant of Maddox.
Female
English
French form of Latin Maria, MARIE means "obstinacy, rebelliousness" or "their rebellion."
Girl/Female
Biblical
Rain, prison.
Female
Welsh
Welsh form of Old French Caterine, CATRIN means "pure."
MATRIX ANALYSIS
MATRIX ANALYSIS
Biblical
coming hither
Boy/Male
Tamil
Welcome rain
Boy/Male
Afghan, Arabic, German, Indian, Iranian, Kurdish, Muslim, Parsi, Zoroastrian
Of Good Birth; Of Noble Family
Female
Native American
Native American Hopi name MANSI means "plucked flower."
Boy/Male
Tamil
God
Boy/Male
Muslim
A loving and caring person to all
Boy/Male
Indian, Sanskrit
Accomplishing Wonderful Deeds
Boy/Male
Gujarati, Hindu, Indian, Sanskrit
Piece of Heart
Girl/Female
Tamil
Kindness, Goddess
Boy/Male
Arabic, Muslim
Angel
MATRIX ANALYSIS
MATRIX ANALYSIS
MATRIX ANALYSIS
MATRIX ANALYSIS
MATRIX ANALYSIS
v. i.
The mineral substance which incloses a vein; a matrix; a gangue.
a.
Of or pertaining to the meter as a standard of measurement; of or pertaining to the decimal system of measurement of which a meter is the unit; as, the metric system; a metric measurement.
v. t.
The white fibrous matter forming the matrix from which fungi.
n.
The five simple colors, black, white, blue, red, and yellow, of which all the rest are composed.
n.
The lifeless portion of tissue, either animal or vegetable, situated between the cells; the intercellular substance.
n.
In type founding and forging, an impression or matrix, formed by a punch drift.
n.
Hence, that which gives form or origin to anything
pl.
of Matrix
n.
A rectangular arrangement of symbols in rows and columns. The symbols may express quantities or operations.
n.
The womb.
a.
Of or pertaining to the Maoris or to their language.
n.
A genus of swallows including the purple martin. See Martin.
n.
The martin.
n.
The cavity in which anything is formed, and which gives it shape; a die; a mold, as for the face of a type.
pl.
of Maori
n.
A mold; a matrix.
n.
A housekeeper; esp., a woman who manages the domestic economy of a public instution; a head nurse in a hospital; as, the matron of a school or hospital.
n.
See Matrix.
n.
The earthy or stony substance in which metallic ores or crystallized minerals are found; the gangue.