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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
Algorithms for matrix decomposition
Web-scale data mining, e.g., see Distributed Nonnegative Matrix Factorization (DNMF), Scalable Nonnegative Matrix Factorization (ScalableNMF), Distributed
Non-negative matrix factorization
Non-negative_matrix_factorization
Matrix used to describe the transitions of a Markov chain
mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing
Stochastic_matrix
Property of a mathematical matrix
positive or zero (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite
Definite_matrix
Square matrix whose off-diagonal entries are nonnegative
In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero): ∀ i ≠ j x i j ≥
Metzler_matrix
In linear algebra, the nonnegative rank of a nonnegative matrix is a concept similar to the usual linear rank of a real matrix, but adding the requirement
Nonnegative rank (linear algebra)
Nonnegative_rank_(linear_algebra)
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 in mathematics
an M-matrix is a matrix whose off-diagonal entries are less than or equal to zero (i.e., it is a Z-matrix) and whose eigenvalues have nonnegative real
M-matrix
Square matrix used to represent a graph or network
computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices
Adjacency_matrix
Matrix representation of a graph
Laplacian is a symmetric matrix if and only if the adjacency matrix A is symmetric and the diagonal entries of D are nonnegative, in which case we can use
Laplacian_matrix
Complex matrix whose conjugate transpose equals its inverse
eiH, where e indicates the matrix exponential, i is the imaginary unit, and H is a Hermitian matrix. For any nonnegative integer n, the set of all n × n
Unitary_matrix
Matrix in linear algebra
linear algebra, a real symmetric matrix A is copositive if x ⊤ A x ≥ 0 {\displaystyle x^{\top }Ax\geq 0} for every nonnegative vector x ≥ 0 {\displaystyle
Copositive_matrix
Type of square matrix
doubly stochastic matrix (also called bistochastic matrix) is a square matrix X = ( x i j ) {\displaystyle X=(x_{ij})} of nonnegative real numbers, each
Doubly_stochastic_matrix
Process of reducing the number of random variables under consideration
3847/1538-4357/aaa1f2. S2CID 3966513. Zhu, Guangtun B. (2016-12-19). "Nonnegative Matrix Factorization (NMF) with Heteroscedastic Uncertainties and Missing
Dimensionality_reduction
Real function with secant line between points above the graph itself
{\displaystyle c} as a nonnegative real number) and an exponential function c e x {\displaystyle ce^{x}} ( c {\displaystyle c} as a nonnegative real number). Convex
Convex_function
A Stieltjes matrix is necessarily an M-matrix. Every n×n Stieltjes matrix is invertible to a nonsingular symmetric nonnegative matrix, though the converse
Stieltjes_matrix
Representation of a matrix as a product
complex, symmetric matrix A. Decomposition: A = V D V T {\displaystyle A=VDV^{\mathsf {T}}} , where D is a real nonnegative diagonal matrix, and V is unitary
Matrix_decomposition
Mathematical operation in linear algebra
eigenvectors are generally different if AB ≠ BA. One may raise a square matrix to any nonnegative integer power multiplying it by itself repeatedly in the same
Matrix_multiplication
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)
Constrained least squares problem
0.CO;2-L. Lin, Chih-Jen (2007). "Projected Gradient Methods for Nonnegative Matrix Factorization" (PDF). Neural Computation. 19 (10): 2756–2779. CiteSeerX 10
Non-negative_least_squares
Dimension of the column space of a matrix
assume that A is an m × n matrix, and we define the linear map f by f(x) = Ax as above. The rank of an m × n matrix is a nonnegative integer and cannot be
Rank_(linear_algebra)
Theorem in linear algebra
nonnegative n-square matrix is the graph with vertices numbered 1, ..., n and arc ij if and only if Aij ≠ 0. If the underlying graph of such a matrix
Perron–Frobenius_theorem
Extracting features from raw data for machine learning
PMID 10548103. Wang, Hua; Nie, Feiping; Huang, Heng; Ding, Chris (2011). "Nonnegative Matrix Tri-factorization Based High-Order Co-clustering and Its Fast Implementation"
Feature_engineering
Matrix equal to its conjugate-transpose
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose
Hermitian_matrix
solution. Any P-matrix is a Q-matrix. Conversely, if a matrix is a Z-matrix and a Q-matrix, then it is also a P-matrix. P-matrix Z-matrix Karamardian, S
Q-matrix
Irreducible representation of the rotation group SO
The d-matrix elements are related to Jacobi polynomials P k ( a , b ) ( cos β ) {\displaystyle P_{k}^{(a,b)}(\cos \beta )} with nonnegative a {\displaystyle
Wigner_D-matrix
Hungarian and American mathematician and physicist (1903–1957)
problem A − λ I q = 0, where the nonnegative matrix A must be square and where the diagonal matrix I is the identity matrix. Von Neumann's irreducibility
John_von_Neumann
Regularization technique for ill-posed problems
Guan, Naiyang; Tao, Dacheng; Luo, Zhigang; Yuan, Bo (2012). "Online nonnegative matrix factorization with robust stochastic approximation". IEEE Transactions
Ridge_regression
matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries
List_of_named_matrices
Most widely known generalized inverse of a matrix
matrix can be decomposed as A = U D V ∗ {\displaystyle A=UDV^{*}} for some isometries U , V {\displaystyle U,V} and diagonal nonnegative real matrix D
Moore–Penrose_inverse
Method of data analysis
doi:10.1086/510127. S2CID 18561804. Zhu, Guangtun B. (2016-12-19). "Nonnegative Matrix Factorization (NMF) with Heteroscedastic Uncertainties and Missing
Principal_component_analysis
Study of matrices and their algebraic properties
|| ... ||, fulfills: Nonnegative: ‖ A ‖ ≥ 0 {\displaystyle \|\mathbf {A} \|\geq 0} with equality only for A = 0, the zero matrix. Scalar multiplication:
Matrix_analysis
Type of matrix in linear algebra
algebra, a square nonnegative matrix A {\displaystyle A} of order n {\displaystyle n} is said to be productive, or to be a Leontief matrix, if there exists
Productive_matrix
Notion in statistics
of nonnegative-definite symmetric matrices in a partially ordered vector space, under the Loewner (Löwner) order. This cone is closed under matrix addition
Fisher_information
Representation of a matrix as a sum
matrix M, M has nonnegative entries, we write M ≥ 0. If M has only positive entries, we write M > 0. Similarly, if the matrix M1 − M2 has nonnegative
Matrix_splitting
Mathematical ring whose elements are matrices
abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set
Matrix_ring
Polynomial with a matrix as variable
determinant of a matrix polynomial with Hermitian positive-definite (semidefinite) coefficients is a polynomial with positive (nonnegative) coefficients
Matrix_polynomial
Mathematical operation
root of a nonnegative integer must either be another integer or an irrational number, excluding non-integer rationals. Contrast that to a matrix of integers
Square_root_of_a_matrix
Tool in linear algebra and matrix analysis
numerical analysis, and statistics. It is defined for a block matrix. Suppose p, q are nonnegative integers such that p + q > 0, and suppose A, B, C, D are
Schur_complement
Branch of applied mathematics
nonnegative matrix A {\displaystyle \mathbf {A} } must be square and where the diagonal matrix I {\displaystyle \mathbf {I} } is the identity matrix.
Mathematical_economics
Separation of a set of source signals from a set of mixed signals
or information-theoretic sense. A second approach, exemplified by nonnegative matrix factorization, is to impose structural constraints on the source signals
Signal_separation
Matrix with one nonzero entry in each row and column
nonsingular matrix and its inverse are both nonnegative matrices (i.e. matrices with nonnegative entries), then the matrix is a generalized permutation matrix. The
Generalized permutation matrix
Generalized_permutation_matrix
Matrices named after Élie Cartan
In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices
Cartan_matrix
Number whose square is a given number
2 = 16 {\displaystyle 4^{2}=(-4)^{2}=16} . Every nonnegative real number x has a unique nonnegative square root, called the principal square root or simply
Square_root
Concepts from linear algebra
(PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used
Eigenvalues_and_eigenvectors
Four-dimensional number system
Lagrange's four-square theorem in number theory, which states that every nonnegative integer is the sum of four integer squares. As well as being an elegant
Quaternion
Arithmetic mean is greater than or equal to geometric mean
geometric mean is similar, except that it is only defined for a list of nonnegative real numbers, and uses multiplication and a root in place of addition
AM–GM_inequality
Square matrix whose off-diagonal entries are nonpositive
definitions, one of which is as follows: a Z-matrix is an M-matrix if it is nonsingular and its inverse is nonnegative. All matrices that are both Z-matrices
Z-matrix_(mathematics)
Complexity of sending information in a distributed algorithm
but also in the theory of nonnegative matrices, where it gives a lower bound on the nonnegative rank of a nonnegative matrix. In the unbounded-error setting
Communication_complexity
Method for analyzing semantic data
IR Research (ECIR-02)", 2002 Chris Ding, Tao Li, Wei Peng (2006). "Nonnegative Matrix Factorization and Probabilistic Latent Semantic Indexing: Equivalence
Probabilistic latent semantic analysis
Probabilistic_latent_semantic_analysis
In mathematics, a totally positive matrix is a square matrix in which all the minors are positive: that is, the determinant of every square submatrix
Totally_positive_matrix
Metzler matrix. A discrete-time linear system x ( k + 1 ) = A x ( k ) {\displaystyle x(k+1)=Ax(k)} is positive if and only if A is a nonnegative matrix. Metzler
Positive_systems
Polish computer scientist (born 1947)
non-negative matrix factorizations and nonnegative tensor decompositions. Moreover, he pioneered in development of multilayer (deep) matrix and tensor factorization
Andrzej_Cichocki
Problem in linear algebra
of the permanent of a matrix is a problem that is thought to be more difficult than the computation of the determinant of a matrix despite the apparent
Computing_the_permanent
Polynomial-time algorithm for the assignment problem
(ckmin(delta, minTo[w])) { wNext = w; } } } // delta will always be nonnegative, // except possibly during the first time this loop runs // if any entries
Hungarian_algorithm
ISBN 978-0-89871-568-2. Cédric Févotte; Nancy Bertin; Jean-Louis Durrieu (2009). "Nonnegative Matrix Factorization with the Itakura-Saito Divergence: With Application
Itakura–Saito_distance
Computer scientist
Chris; He, Xiaofeng; Simon, Horst D (2005). "On the Equivalence of Nonnegative Matrix Factorization and Spectral Clustering". Proceedings of the 2005 SIAM
Horst_D._Simon
Matrix that converges to zero matrix
have only nonnegative entries. If the splitting (5) is a regular splitting of the matrix A and A−1 ≥ 0, then ρ(T) < 1 and T is a convergent matrix. Hence
Convergent_matrix
Set of computational techniques
PMC 7181686. PMID 32332754. Tang D, Park S, Zhao H (March 2020). "NITUMID: Nonnegative matrix factorization-based Immune-TUmor MIcroenvironment Deconvolution".
Cellular_deconvolution
American mathematician
A. (1966). "On the permanent and maximal characteristic root of a nonnegative matrix". Proc. Amer. Math. Soc. 17 (6): 1413–1416. doi:10.1090/s0002-9939-1966-0204444-2
Richard_A._Brualdi
Every square matrix with positive entries can be written in a certain standard form
theorem states that every square matrix with positive entries can be written in a certain standard form. If A is an n × n matrix with strictly positive elements
Sinkhorn's_theorem
Australian and American mathematician (born 1975)
the nonpositivity of which is equivalent to the Riemann hypothesis, is nonnegative. In 2020, Tao proved Sendov's conjecture, concerning the locations of
Terence_Tao
Estimates values in an N-dimensional matrix
following concept: Z {\displaystyle Z} , matrix Y {\displaystyle Y} and matrix X {\displaystyle X} are known real nonnegative matrices of dimension n , m {\displaystyle
Iterative proportional fitting
Iterative_proportional_fitting
Chernoff bounds concern the sum of independent, nonnegative, and uniformly bounded random variables. In the matrix setting, the analogous theorem concerns a
Matrix_Chernoff_bound
Arithmetic operations
{\displaystyle 2\times 7=\min\{2,7\}=2.} The lunar arithmetic operations on nonnegative multidigit numbers are performed as in usual arithmetic as illustrated
Lunar_arithmetic
Polynomial of the elements of a matrix
polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries", Journal of the ACM, 51 (4): 671–697, CiteSeerX 10.1.1
Permanent_(mathematics)
Technique in numerical linear algebra
which case the approximation is nonnegative. Computer algebra, in which case the approximation is Sylvester structured. Matrix product states, in which case
Low-rank_approximation
Bound on eigenvalues
entries are nonnegative Muirhead's inequality Bendixson's inequality Schur–Horn theorem – Characterizes the diagonal of a Hermitian matrix with given eigenvalues
Gershgorin_circle_theorem
Matrix used to analyze the stability of a polynomial by its coefficients
Topics in matrix analysis. Cambridge University Press. p. 101. ISBN 0-521-30587-X. Asner, Bernard A. Jr. (1970). "On the Total Nonnegativity of the Hurwitz
Routh–Hurwitz_matrix
Probability concept
distribution λ {\displaystyle \lambda } and rate matrix Q {\displaystyle Q} to mean: for any nonnegative integer n ≥ 0 {\displaystyle n\geq 0} , for all
Continuous-time_Markov_chain
Criterion for positivity of a Hermitian matrix
} A Hermitian matrix M is positive-semidefinite if and only if all principal minors of M are nonnegative. Suppose M n {\displaystyle M_{n}}
Sylvester's_criterion
American mathematician (1943–2024)
was a consequence of a stronger inequality which asserts the nonnegativity of a matrix-valued function.[H93a] His result required the stronger assumption
Richard_S._Hamilton
Sub-field of genomics
context-dependent mutations can be represented in the form of a nonnegative matrix. This matrix can be further decomposed into components (mutational signatures)
Oncogenomics
[ a i j ] {\displaystyle A=[a_{ij}]} be a square matrix of order n {\displaystyle n} with nonnegative entries and with ∑ i = 1 n ( ∑ j = 1 n a i j ) =
Dittert_conjecture
Multiplicative factor in a mathematical expression
1 + a 0 {\displaystyle a_{k}x^{k}+\dotsb +a_{1}x^{1}+a_{0}} for some nonnegative integer k {\displaystyle k} , where a k , … , a 1 , a 0 {\displaystyle
Coefficient
South Korean American mathematician
(2008-01-01). "Nonnegative Matrix Factorization Based on Alternating Nonnegativity Constrained Least Squares and Active Set Method". SIAM Journal on Matrix Analysis
Haesun_Park
Formula for the "volume" of an n-simplex
determinant of the volume matrix being positive. Let K be a positive integer and D be a 1n × n symmetric hollow matrix with nonnegative elements, with n ≥ 2
Cayley–Menger_determinant
the inverses b − 1 ≤ a − 1 {\displaystyle b^{-1}\leq a^{-1}} holds. Nonnegative matrix Positive operator (Hilbert space) Palmer 2001, p. 798. Blackadar 2006
Positive_element
Space formed by the ''n''-tuples of real numbers
finite sequences of nonnegative numbers), and a convex set is an algebra over the universal simplex (of finite sequences of nonnegative numbers summing to
Real_coordinate_space
Jun; Yang, Gang; Du, Bo; Zhang, Liangpei (June 2017). "A Poisson nonnegative matrix factorization method with parameter subspace clustering constraint
Land_cover_maps
Expression for sums of powers
of finding polynomials S h , d p {\displaystyle S_{h,d}^{p}} for any nonnegative integer p {\displaystyle p} such that S h , d p ( n ) = ∑ k = 0 n − 1
Faulhaber's_formula
Computational methods in biology
(2018-07-24). "Integrative analysis of single-cell genomics data by coupled nonnegative matrix factorizations". Proceedings of the National Academy of Sciences.
Single-cell multi-omics integration
Single-cell_multi-omics_integration
⋯ ∣ a m {\displaystyle a_{1}\mid a_{2}\mid \cdots \mid a_{m}} . The nonnegative integer r {\displaystyle r} is called the free rank or Betti number of
Invariant_factor
Mathematical inequality
matrix P is doubly stochastic precisely if both P and its transpose PT are stochastic matrices. A stochastic matrix is a square matrix of nonnegative
Muirhead's_inequality
Pictorial representation of symmetry
and paracompact if all its subgroups are finite or affine (i.e. have nonnegative determinants). Finite and affine groups are also called elliptical and
Coxeter–Dynkin_diagram
Theorem in graph theory
vertex. In this case the matrix is interpreted as the adjacency matrix of such a directed graph. When are pairs of nonnegative integers ( ( a 1 , b 1 )
Gale–Ryser_theorem
Doubly stochastic matrix
A square matrix B of size n is doubly stochastic (or bistochastic) if all its rows and columns sum to 1 and all its entries are nonnegative real numbers
Orthostochastic_matrix
Formula in Lie theory
\prod _{i=1}^{n}r_{i}!s_{i}!}},} where the sum is performed over all nonnegative values of s i {\displaystyle s_{i}} and r i {\displaystyle r_{i}} , and
Baker–Campbell–Hausdorff formula
Baker–Campbell–Hausdorff_formula
Parameter estimation technique in statistics, particularly econometrics
To conduct the test we compute the value of J from the data. It is a nonnegative number. We compare it with (for example) the 0.95 quantile of the χ k
Generalized_method_of_moments
Property of a function
superadditive for nonnegative Hermitian matrix, that is, if A , B ∈ Mat n ( C ) {\displaystyle A,B\in {\text{Mat}}_{n}(\mathbb {C} )} are nonnegative Hermitian
Superadditivity
Indian mathematician
Algebra and Linear Models. Springer. Bapat, RB, T. E. S. Raghavan (2009). Nonnegative Matrices and Applications. Cambridge University Press.{{cite book}}:
Ravindra_Bapat
Concept in quantum physics
density matrix describing knowledge of the state of a subsystem. The entropy of entanglement is the Von Neumann entropy of the reduced density matrix for
Entropy_of_entanglement
Generalisation of Fourier transform to any ring
a^{(q'-1)/4}\}} , in exact analogy with the complex case. They occur with some nonnegative multiplicity. The number-theoretic transform (NTT) is obtained by specializing
Discrete Fourier transform over a ring
Discrete_Fourier_transform_over_a_ring
testable sufficient conditions for a form to be SOS. Moreover, every real nonnegative form can be approximated as closely as desired (in the l 1 {\displaystyle
Polynomial_SOS
Matrix form in linear algebra
the left of pivots are nonnegative and strictly smaller than the pivot. Note that the row-style definition has a unimodular matrix U {\displaystyle U} multiplying
Hermite_normal_form
Method for estimating the unknown parameters in a linear regression model
{\hat {\beta }}\mid X\,]\geq 0} in the sense that this is a nonnegative-definite matrix. This theorem establishes optimality only in the class of linear
Ordinary_least_squares
American mathematician (born 1948)
"Review of Matrix Analysis, 2nd edition". MAA Reviews, Mathematical Association of America. Garloff, Jürgen (2012). "Review of Totally Nonnegative Matrices
Charles_Royal_Johnson
Number of subsets of a given size
can be extended to accept more general families of inputs. When n is a nonnegative integer and k is an integer such that k < 0 or k > n, it is common to
Binomial_coefficient
True when either but not both inputs are true
) ∧ ¬ ( p ∧ q ) {\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\vee q)\land \neg (p\land q)\end{matrix}}} The exclusive disjunction p ↮ q {\displaystyle
Exclusive_or
Mathematical algorithm
finding a minimum of a non-linear function. Since a sum of squares must be nonnegative, the algorithm can be viewed as using Newton's method to iteratively
Gauss–Newton_algorithm
NONNEGATIVE MATRIX
NONNEGATIVE MATRIX
NONNEGATIVE MATRIX
NONNEGATIVE MATRIX
Female
English
(Δήλια) Greek name DELIA means "of Delos." In mythology, this is a name borne by Artemis, referring to her place of birth.
Girl/Female
Bengali, Hindu, Indian, Tamil, Telugu
Logically Intelligent; Who Stands Alone
Girl/Female
Hindu, Indian, Kannada, Tamil
Brightness of the Rising Sun
Boy/Male
Hindu, Indian, Marathi
Season; Hot and Windy Season
Male
Hebrew
(×ַבְרָ×) Variant spelling of Hebrew Abram, AVRAM means "father of elevation."Â
Surname or Lastname
English
English : metonymic occupational name for a basket weaver, from Anglo-Norman French banastre ‘basket’ (the result of a Late Latin cross between Gaulish benna and Greek kanistron). The term denoting a stair rail is unconnected with this name; it was not used before the 17th century.
Girl/Female
Hindu
Princess
Female
Norwegian
Norwegian variant spelling of Scandinavian Gunvor, GUNNVOR means "war-vigilant."Â
Boy/Male
Hindu, Indian, Kannada, Malayalam, Marathi, Sanskrit, Telugu
King of Fragrance
Girl/Female
Arabic, Muslim
A Distinguished Woman of her Times was so Named and She was the Daughter of Saad Al-aslamiyah and She Offered Allegiance (Bayah) to the Prophet PBUH
NONNEGATIVE MATRIX
NONNEGATIVE MATRIX
NONNEGATIVE MATRIX
NONNEGATIVE MATRIX
NONNEGATIVE MATRIX
n.
The matrix, or cavity, in which anything is shaped, and from which it takes its form; also, the body or mass containing the cavity; as, a sand mold; a jelly mold.
n.
The amorphous or homogenous matrix or ground mass, as distinguished from well-defined crystals; as, the magma of porphyry.
n.
A mold; a matrix.
n.
A protoplasmic animal cell; esp., such as float free, like blood, lymph, and pus corpuscles; or such as are imbedded in an intercellular matrix, like connective tissue and cartilage corpuscles. See Blood.
n.
The five simple colors, black, white, blue, red, and yellow, of which all the rest are composed.
n.
The womb.
n.
Hence, that which gives form or origin to anything
pl.
of Matrix
n.
A cutting or engraving; a figure cut into something, as a gem, so as to make a design depressed below the surface of the material; hence, anything so carved or impressed, as a gem, matrix, etc.; -- opposed to cameo. Also used adjectively.
n.
See Matrix.
n.
A kind of cartilage with a fibrous matrix and approaching fibrous connective tissue in structure.
n.
The earthy or stony substance in which metallic ores or crystallized minerals are found; the gangue.
n.
A mold or matrix in which anything is cast or formed to a particular shape.
n.
A rectangular arrangement of symbols in rows and columns. The symbols may express quantities or operations.
n.
The lifeless portion of tissue, either animal or vegetable, situated between the cells; the intercellular substance.
v. i.
The mineral substance which incloses a vein; a matrix; a gangue.
n.
In type founding and forging, an impression or matrix, formed by a punch drift.
n.
One of the protoplasmic cells which occur in the osteogenetic layer of the periosteum, and from or around which the matrix of the bone is developed; an osteoplast.
n.
The cavity in which anything is formed, and which gives it shape; a die; a mold, as for the face of a type.
v. t.
The white fibrous matter forming the matrix from which fungi.