Cyclic Subspaces, Duality and the Jordan Canonical Form View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

2015

AUTHORS

Jörg Liesen , Volker Mehrmann

ABSTRACT

In this chapter we use the duality theory to analyze the properties of an endomorphism f on a finite dimensional vector space \(\mathcal V\) in detail. We are particularly interested in the algebraic and geometric multiplicities of the eigenvalues of f and the characterization of the corresponding eigenspaces. Our strategy in this analysis is to decompose the vector space \(\mathcal V\) into a direct sum of f-invariant subspaces so that, with appropriately chosen bases, the essential properties of f will be obvious from its matrix representation. The matrix representation that we derive is called the Jordan canonical form of f. Because of its great importance there have been many different derivations of this form using different mathematical tools. More... »

PAGES

227-251

Book

TITLE

Linear Algebra

ISBN

978-3-319-24344-3
978-3-319-24346-7

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-319-24346-7_16

DOI

http://dx.doi.org/10.1007/978-3-319-24346-7_16

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1000969085


Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
Incoming Citations Browse incoming citations for this publication using opencitations.net

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

[
  {
    "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
    "about": [
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Pure Mathematics", 
        "type": "DefinedTerm"
      }, 
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Mathematical Sciences", 
        "type": "DefinedTerm"
      }
    ], 
    "author": [
      {
        "affiliation": {
          "name": [
            "Institute of Mathematics, Technical University of Berlin"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Liesen", 
        "givenName": "J\u00f6rg", 
        "type": "Person"
      }, 
      {
        "affiliation": {
          "name": [
            "Institute of Mathematics, Technical University of Berlin"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Mehrmann", 
        "givenName": "Volker", 
        "type": "Person"
      }
    ], 
    "datePublished": "2015", 
    "datePublishedReg": "2015-01-01", 
    "description": "In this chapter we use the duality theory to analyze the properties of an endomorphism f on a finite dimensional vector space\u00a0\\(\\mathcal V\\) in detail. We are particularly interested in the algebraic and geometric multiplicities of the eigenvalues of f and the characterization of the corresponding eigenspaces. Our strategy in this analysis is to decompose the vector space \\(\\mathcal V\\) into a direct sum of f-invariant subspaces so that, with appropriately chosen bases, the essential properties of f will be obvious from its matrix representation. The matrix representation that we derive is called the Jordan canonical form of f. Because of its great importance there have been many different derivations of this form using different mathematical tools.", 
    "genre": "chapter", 
    "id": "sg:pub.10.1007/978-3-319-24346-7_16", 
    "inLanguage": [
      "en"
    ], 
    "isAccessibleForFree": false, 
    "isPartOf": {
      "isbn": [
        "978-3-319-24344-3", 
        "978-3-319-24346-7"
      ], 
      "name": "Linear Algebra", 
      "type": "Book"
    }, 
    "name": "Cyclic Subspaces, Duality and the Jordan Canonical Form", 
    "pagination": "227-251", 
    "productId": [
      {
        "name": "doi", 
        "type": "PropertyValue", 
        "value": [
          "10.1007/978-3-319-24346-7_16"
        ]
      }, 
      {
        "name": "readcube_id", 
        "type": "PropertyValue", 
        "value": [
          "049e1db66ac2ab41cae8085f58d66fd13275238190160d0363db9f740500bae6"
        ]
      }, 
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "pub.1000969085"
        ]
      }
    ], 
    "publisher": {
      "location": "Cham", 
      "name": "Springer International Publishing", 
      "type": "Organisation"
    }, 
    "sameAs": [
      "https://doi.org/10.1007/978-3-319-24346-7_16", 
      "https://app.dimensions.ai/details/publication/pub.1000969085"
    ], 
    "sdDataset": "chapters", 
    "sdDatePublished": "2019-04-15T10:16", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com-uberresearch-data-dimensions-target-20181106-alternative/cleanup/v134/2549eaecd7973599484d7c17b260dba0a4ecb94b/merge/v9/a6c9fde33151104705d4d7ff012ea9563521a3ce/jats-lookup/v90/0000000001_0000000264/records_8659_00000001.jsonl", 
    "type": "Chapter", 
    "url": "http://link.springer.com/10.1007/978-3-319-24346-7_16"
  }
]
 

Download the RDF metadata as:  json-ld nt turtle xml License info

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/978-3-319-24346-7_16'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/978-3-319-24346-7_16'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/978-3-319-24346-7_16'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/978-3-319-24346-7_16'


 

This table displays all metadata directly associated to this object as RDF triples.

65 TRIPLES      21 PREDICATES      26 URIs      19 LITERALS      7 BLANK NODES

Subject Predicate Object
1 sg:pub.10.1007/978-3-319-24346-7_16 schema:about anzsrc-for:01
2 anzsrc-for:0101
3 schema:author Ndbc9a1ab9ad94d0b8e4de656b71d1c87
4 schema:datePublished 2015
5 schema:datePublishedReg 2015-01-01
6 schema:description In this chapter we use the duality theory to analyze the properties of an endomorphism f on a finite dimensional vector space \(\mathcal V\) in detail. We are particularly interested in the algebraic and geometric multiplicities of the eigenvalues of f and the characterization of the corresponding eigenspaces. Our strategy in this analysis is to decompose the vector space \(\mathcal V\) into a direct sum of f-invariant subspaces so that, with appropriately chosen bases, the essential properties of f will be obvious from its matrix representation. The matrix representation that we derive is called the Jordan canonical form of f. Because of its great importance there have been many different derivations of this form using different mathematical tools.
7 schema:genre chapter
8 schema:inLanguage en
9 schema:isAccessibleForFree false
10 schema:isPartOf Nc48d10a711c74824a978e9f7f266852b
11 schema:name Cyclic Subspaces, Duality and the Jordan Canonical Form
12 schema:pagination 227-251
13 schema:productId N23224ad35512453fa73301a670e1655a
14 Ncc54412625f5412ba1c40e334f17eb9a
15 Nfeead261c163495e877a46671b2ab685
16 schema:publisher N640babff9af4407fb59dabcb8387f158
17 schema:sameAs https://app.dimensions.ai/details/publication/pub.1000969085
18 https://doi.org/10.1007/978-3-319-24346-7_16
19 schema:sdDatePublished 2019-04-15T10:16
20 schema:sdLicense https://scigraph.springernature.com/explorer/license/
21 schema:sdPublisher N15ee570da6604687a2bce9844fd99047
22 schema:url http://link.springer.com/10.1007/978-3-319-24346-7_16
23 sgo:license sg:explorer/license/
24 sgo:sdDataset chapters
25 rdf:type schema:Chapter
26 N15ee570da6604687a2bce9844fd99047 schema:name Springer Nature - SN SciGraph project
27 rdf:type schema:Organization
28 N23224ad35512453fa73301a670e1655a schema:name readcube_id
29 schema:value 049e1db66ac2ab41cae8085f58d66fd13275238190160d0363db9f740500bae6
30 rdf:type schema:PropertyValue
31 N6133426788554757814e9a66b3a2fe82 schema:affiliation Nb8ca600f34984e2ea2529a5e34081e7b
32 schema:familyName Liesen
33 schema:givenName Jörg
34 rdf:type schema:Person
35 N640babff9af4407fb59dabcb8387f158 schema:location Cham
36 schema:name Springer International Publishing
37 rdf:type schema:Organisation
38 N69ef0849709d4662920e67d4405a960d schema:name Institute of Mathematics, Technical University of Berlin
39 rdf:type schema:Organization
40 N85e1b50f9de6481d9655b4a003702322 schema:affiliation N69ef0849709d4662920e67d4405a960d
41 schema:familyName Mehrmann
42 schema:givenName Volker
43 rdf:type schema:Person
44 Nb8ca600f34984e2ea2529a5e34081e7b schema:name Institute of Mathematics, Technical University of Berlin
45 rdf:type schema:Organization
46 Nc48d10a711c74824a978e9f7f266852b schema:isbn 978-3-319-24344-3
47 978-3-319-24346-7
48 schema:name Linear Algebra
49 rdf:type schema:Book
50 Ncb8eca0af8944373bf401a03d1aa7014 rdf:first N85e1b50f9de6481d9655b4a003702322
51 rdf:rest rdf:nil
52 Ncc54412625f5412ba1c40e334f17eb9a schema:name doi
53 schema:value 10.1007/978-3-319-24346-7_16
54 rdf:type schema:PropertyValue
55 Ndbc9a1ab9ad94d0b8e4de656b71d1c87 rdf:first N6133426788554757814e9a66b3a2fe82
56 rdf:rest Ncb8eca0af8944373bf401a03d1aa7014
57 Nfeead261c163495e877a46671b2ab685 schema:name dimensions_id
58 schema:value pub.1000969085
59 rdf:type schema:PropertyValue
60 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
61 schema:name Mathematical Sciences
62 rdf:type schema:DefinedTerm
63 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
64 schema:name Pure Mathematics
65 rdf:type schema:DefinedTerm
 




Preview window. Press ESC to close (or click here)


...