Zero matrix

In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of $m\times n$ matrices, and is denoted by the symbol $O$ or $0$ followed by subscripts corresponding to the dimension of the matrix as the context sees fit.^[1]^[2]^[3] Some examples of zero matrices are

0_{1,1}={\begin{bmatrix}0\end{bmatrix}},\ 0_{2,2}={\begin{bmatrix}0&0\\0&0\end{bmatrix}},\ 0_{2,3}={\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}}.\

Properties

The set of $m\times n$ matrices with entries in a ring K forms a ring $K_{m,n}$ . The zero matrix $0_{K_{m,n}}\,$ in $K_{m,n}\,$ is the matrix with all entries equal to $0_{K}\,$ , where $0_{K}$ is the additive identity in K.

0_{K_{m,n}}={\begin{bmatrix}0_{K}&0_{K}&\cdots &0_{K}\\0_{K}&0_{K}&\cdots &0_{K}\\\vdots &\vdots &\ddots &\vdots \\0_{K}&0_{K}&\cdots &0_{K}\end{bmatrix}}_{m\times n}

The zero matrix is the additive identity in $K_{m,n}\,$ .^[4] That is, for all $A\in K_{m,n}\,$ it satisfies the equation

0_{K_{m,n}}+A=A+0_{K_{m,n}}=A.

There is exactly one zero matrix of any given dimension m×n (with entries from a given ring), so when the context is clear, one often refers to the zero matrix. In general, the zero element of a ring is unique, and is typically denoted by 0 without any subscript indicating the parent ring. Hence the examples above represent zero matrices over any ring.

The zero matrix also represents the linear transformation which sends all the vectors to the zero vector.^[5] It is idempotent, meaning that when it is multiplied by itself, the result is itself.

The zero matrix is the only matrix whose rank is 0.

Occurrences

In ordinary least squares regression, if there is a perfect fit to the data, the annihilator matrix is the zero matrix.

References

^ Lang, Serge (1987), Linear Algebra, Undergraduate Texts in Mathematics, Springer, p. 25, ISBN 9780387964126, We have a zero matrix in which a_ij = 0 for all i, j. ... We shall write it O.
^ "Intro to zero matrices (article) | Matrices". Khan Academy. Retrieved 2020-08-13.
^ Weisstein, Eric W. "Zero Matrix". mathworld.wolfram.com. Retrieved 2020-08-13.
^ Warner, Seth (1990), Modern Algebra, Courier Dover Publications, p. 291, ISBN 9780486663418, The neutral element for addition is called the zero matrix, for all of its entries are zero.
^ Bronson, Richard; Costa, Gabriel B. (2007), Linear Algebra: An Introduction, Academic Press, p. 377, ISBN 9780120887842, The zero matrix represents the zero transformation 0, having the property 0(v) = 0 for every vector v ∈ V.

v t e Matrix classes
Explicitly constrained entries	Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz-stable Positive-definite Stieltjes
Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Routh-Hurwitz Seifert Shear Similarity Symplectic Totally positive Transformation
Used in statistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
Mathematics portal List of matrices Category:Matrices

Read other articles:

Acerno

Concattedrale Acerno Acerno adalah kota (komune) di region Campania di provinsi salerno, Italia. Kota ini terletak di sebelah tenggara dari Ibu kota Roma. Geografi Acerno memiliki populasi 2.867 jiwa pada akhir Maret 2009[1] adalah sebuah kota kecil, atau desa yang besar, terletak 40 km sebelah utara-timur dari ibu kota provinsi Salerno pada 727 meter di atas permukaan laut di lembah Tusciano sebuah sungai yang naik di lereng Monte Polveracchio . Ini terletak dalam Parco regional...

Orientasi romantis

Orientasi seksual Berbagai orientasi seksual Aseksual Biseksual Heteroseksual Homoseksual Kategori non-biner Androfilia dan ginefilia Aseksualitas abu-abu Nonheteroseksual Panseksualitas Poliseksualitas Queer Penelitian Biologi Demografi Ilmu saraf Ilmu kedokteran Kisi Klein Kontinum Lingkungan Orientasi romantis Penelitian queer Seksologi Seksualitas perempuan Seksualitas laki-laki Skala Kinsey Pada hewan: Perilaku homoseksual pada hewan (Daftar) Kategorilbs Orientasi romantis, disebut juga ...

Howard T. Odum

American ecologist (1924–2002) This article is about the American ecologist. For the American sociologist, see Howard W. Odum. This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages) This article's lead section may be too short to adequately summarize the key points. Please consider expanding the lead to provide an accessible overview of all important aspects of the article. (April 2021)This ar...

Straton Lampsakenos

Straton LampsakenosStraton, digambarkan sebagai sarjana abad pertengahan di Kronik NurembergLahirskt. 335 SMLampsakusMeninggalskt. 269 SMAthenaEraFilsafat KunoKawasanFilosofi BaratAliranSekolah PeripatetikMinat utamaFilsafat alam, Fisika Dipengaruhi Aristoteles, Theophrastus Memengaruhi Pierre Bayle Straton Lampsakenos (/ˈstreɪtoʊ/; Yunani: Στράτων ὁ Λαμψακηνός, Straton ho Lampsakenos, skt. 335 – skt. 269 SM) merupakan seorang filsuf Peripatetic, dan direktur...

Kerajaan Shule

Kerajaan Shule疏勒+ 200 SM–790Cekungan Tarim pada abad ke-3 M. Daerah Kerajaan Shule berwarna ungu.Ibu kotaKashgarBahasa yang umum digunakanKanchak (dialek bahasa Saka)PemerintahanMonarki• ?-73 AD Cheng• 168-170 Hede Sejarah • Didirikan + 200 SM• Dikuasai oleh Tibet 790 Sunting kotak info • Lihat • BicaraBantuan penggunaan templat ini Kerajaan Shule (Hanzi: 疏勒) dahulu adalah sebuah kerajaan di wilayah oasis di Gurun Taklamakan di...

Fenella Fielding

English actress (1927–2018) Fenella FieldingOBEFenella Fielding on her 90th birthday in 2017BornFenella Marion Feldman(1927-11-17)17 November 1927Hackney, London, EnglandDied11 September 2018(2018-09-11) (aged 90)Hammersmith, London, EnglandOccupationActressYears active1952–2018RelativesBasil Feldman, Baron Feldman (brother)Nick Feldman (nephew)Websitehttp://www.fenellafielding.com/ Fenella Fielding, OBE (born Fenella Marion Feldman; 17 November 1927 – 11 September 2018)&#...

Raymond Franz

Raymond FranzFranz, awal 1980-anLahirRaymond Victor Franz(1922-05-08)8 Mei 1922Cincinnati, Ohio, Amerika SerikatMeninggal2 Juni 2010(2010-06-02) (umur 88)Winston, Georgia, Amerika SerikatPekerjaanPengkhotbahSuami/istriCynthia Badame Bagian dari seriSaksi-Saksi Yehuwa Ikhtisar Struktur organisasi Badan Pimpinan Watch Tower Bibleand Tract Society Badan usaha Sejarah Gerakan Siswa Alkitab Sengketa kepemimpinan Kelompok-kelompok pecahan Perkembangan doktrin Prediksi-prediksi keliru Demograf...

2020年夏季奥林匹克运动会马来西亚代表团

2020年夏季奥林匹克运动会马来西亚代表團马来西亚国旗IOC編碼MASNOC马来西亚奥林匹克理事会網站olympic.org.my（英文）2020年夏季奥林匹克运动会（東京）2021年7月23日至8月8日（受2019冠状病毒病疫情影响推迟，但仍保留原定名称）運動員30參賽項目10个大项旗手开幕式：李梓嘉和吳柳螢（羽毛球）[1][2]閉幕式：潘德莉拉（跳水）[3]獎牌榜排名第74 金牌銀牌銅�...

Brian Wilson Presents Smile

2004 studio album by Brian Wilson This article is about the rerecorded 2004 version of Smile. For the unfinished album, see Smile (The Beach Boys album). Brian Wilson Presents SmileStudio album by Brian WilsonReleasedSeptember 28, 2004 (2004-09-28)RecordedApril 13 – July 2004StudioSunset Sound Recorders and Your Place or Mine Recording, CaliforniaGenreOrchestral pop[1][2]art pop[3]psychedelic pop[4]Americana[5]indie rock[6]L...

Філософія

Частина серії проФілософіяLeft to right: Plato, Kant, Nietzsche, Buddha, Confucius, AverroesПлатонКантНіцшеБуддаКонфуційАверроес Філософи Епістемологи Естетики Етики Логіки Метафізики Соціально-політичні філософи Традиції Аналітична Арістотелівська Африканська Близькосхідна іранська Буддій�...

District de Changqing

Cet article est une ébauche concernant la Chine. Vous pouvez partager vos connaissances en l’améliorant (comment ?) selon les recommandations des projets correspondants. Chángqīng Qū 长清区 Administration Pays Chine Province ou région autonome Shandong Préfecture Jinan Statut administratif District Code postal 250300[1] Indicatif +86 (0) Démographie 571 900 hab. (2019) Densité 473 hab./km2 Géographie Coordonnées 36° 26′ 57″ nord, 116°&...

André Marie Jean Jacques Dupin

French politician, magistrate and lawyer (1783–1865) This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources.Find sources: André Marie Jean Jacques Dupin – news · newspapers · books · scholar · JSTOR (February 2012) André Marie Jean Jacques DupinPresident of the Chamber of DeputiesIn office29 April 1832 – ...

Hyderabad railway division

Railway division of India This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Hyderabad railway division – news · newspapers · books · scholar · JSTOR (November 2015) (Learn how and when to remove this message) Hyderabad railway divisionHeadquarters at KachegudaOverviewHeadquartersKachegudaReporting markHYBLocal...

Triton (Mythologie)

Triton trägt Theseus (Melisches Relief, 5. Jh. v. Chr.) Triton und Nereide (Neapel) Triton trägt Nymphe (Berlin) Triton (altgriechisch Τρίτων Trítōn, neugriechisch Τρίτωνας Trítonas) ist ein Meeresgott der griechischen Mythologie, von dem später die mythologische Gattung der Tritonen abgeleitet wurde. Inhaltsverzeichnis 1 Mythologie 2 Tritonen 3 Rezeption 4 Literatur 5 Anmerkungen 6 Weblinks Mythologie Triton wurde als Mischwesen gedacht und oft als Sohn des Poseidon und d...

Ш-2

Не следует путать с Ш2, студией звукозаписи. Ш-2 Ш-2 в Монино Тип самолёт-амфибия Разработчик ОКБ Шаврова Производитель Завод № 31 (Таганрог) Завод № 23 (Ленинград) Главный конструктор В. Б. Шавров Первый полёт 11 ноября 1930 года Начало эксплуатации 1932 год Конец эксплуатации 1964 г...

2012 Republic of Macedonia inter-ethnic violence

Ethnic conflict between Albanians and Macedonians in 2012 This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages) This article possibly contains original research. Please improve it by verifying the claims made and adding inline citations. Statements consisting only of original research should be removed. (April 2015) (Learn how and when to remove this message) This article relies excessively on ...

Al Wakrah

This article is about the city in eastern Qatar. For the municipality in which the city is located, see Al Wakrah (municipality). City in Al Wakrah Municipality, QatarAl Wakrah الوكرةCityTop to Bottom, Left to Right: A lone fishing vessel on the coastline, Workers traveling along Al Wakrah Corniche in the Al Wakrah Heritage Village, Traditional stone buildings in Al Wakrah Heritage Village, Renovated fort of Sheikh Abdulrahman bin Jassim Al Thani (now known as Al Wakrah Fort) SealAl Wak...

Elena Ducas

Elena Angelina Ducaina (1242 – Nuceria Christianorum, prima dell'11 marzo 1271) è stata una principessa bizantina, regina consorte di Sicilia come moglie di Manfredi di Sicilia. Elena Angelina DucainaRegina consorte di SiciliaIn carica2 giugno 1259 –26 febbraio 1266 PredecessoreBeatrice di Savoia SuccessoreBeatrice di Provenza Nascita1242 MorteNuceria Christianorum, 14 marzo 1271 DinastiaAngelo Comneno Ducas (nascita)Hohenstaufen (matrimonio) PadreMichele II d'Epiro MadreTeodo...

Fundación Fútbol Club Barcelona

Fundación Fútbol Club Barcelona Tipo fundaciónFundación 1994Fundador Fútbol Club BarcelonaSede central Barcelona, EspañaÁrea de operación Ámbito mundialDeporte fútbolSitio web FundacionFCB.com[editar datos en Wikidata] La Fundación Fútbol Club Barcelona es una entidad social, humanitaria, cultural y deportiva creada en 1994 para promover e incentivar el trabajo por aquellas comunidades infantiles y adultas que se consideran vulnerables. Historia Inicialmente las activida...

IRGs

Immunity Related Guanosine Triphosphatases or IRGs are proteins activated as part of an early immune response. IRGs have been described in various mammals but are most well characterized in mice. IRG activation in most cases is induced by an immune response and leads to clearance of certain pathogens. Figure 1: Crystal structure of a Mouse IRG (PDB id 1TQ6) Background Interferon (IFN)-inducible GTPases encompass four families of proteins including myxovirus resistant proteins (Mx), guanylate...

PROFILPELAJAR.COM