Upper triangular matrix - Revision history

Vipul: /* Stronger properties */

2014-05-01T17:54:57Z

Stronger properties

← Older revision		Revision as of 17:54, 1 May 2014
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	\| [[Weaker than::scalar matrix]] \|\| scalar multiple of [[identity matrix]] \|\| (via diagonal matrix) \|\| (via diagonal matrix) \|\| {{intermediate notions short\|upper triangular matrix\|scalar matrix}}		\| [[Weaker than::scalar matrix]] \|\| scalar multiple of [[identity matrix]] \|\| (via diagonal matrix) \|\| (via diagonal matrix) \|\| {{intermediate notions short\|upper triangular matrix\|scalar matrix}}
	\|-		\|-
	\| [[Weaker than::diagonal matrix]] \|\| all entries not on the main diagonal are zero \|\| \|\| the matrix <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\\end{pmatrix}</math> is a counterexample \|\| {{intermediate notions short\|upper triangular matrix\|diagonal matrix}}		\| [[Weaker than::diagonal matrix]] \|\| all entries not on the main diagonal are zero \|\| \|\| the matrix <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}</math> is a counterexample \|\| {{intermediate notions short\|upper triangular matrix\|diagonal matrix}}
	\|-		\|-
	\| [[Weaker than::upper unitriangular matrix]] \|\| upper triangular matrix with all diagonal entries equal to 1 \|\| \|\| \|\| {{intermediate notions short\|upper triangular matrix\|upper unitriangular matrix}}		\| [[Weaker than::upper unitriangular matrix]] \|\| upper triangular matrix with all diagonal entries equal to 1 \|\| \|\| \|\| {{intermediate notions short\|upper triangular matrix\|upper unitriangular matrix}}
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	\| [[Weaker than::strictly upper triangular matrix]] \|\| all entries on ''and'' below the main diagonal are zero \|\| \|\| \|\| {{intermediate notions short\|upper triangular matrix\|strictly upper triangular matrix}}		\| [[Weaker than::strictly upper triangular matrix]] \|\| all entries on ''and'' below the main diagonal are zero \|\| \|\| \|\| {{intermediate notions short\|upper triangular matrix\|strictly upper triangular matrix}}
	\|}		\|}

	===Contrast properties===		===Contrast properties===

Vipul: /* Contrast properties= */

2014-05-01T17:54:41Z

Contrast properties=

← Older revision		Revision as of 17:54, 1 May 2014
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	\| [[Weaker than::strictly upper triangular matrix]] \|\| all entries on ''and'' below the main diagonal are zero \|\| \|\| \|\| {{intermediate notions short\|upper triangular matrix\|strictly upper triangular matrix}}		\| [[Weaker than::strictly upper triangular matrix]] \|\| all entries on ''and'' below the main diagonal are zero \|\| \|\| \|\| {{intermediate notions short\|upper triangular matrix\|strictly upper triangular matrix}}
	\|}		\|}
	==Contrast properties===		===Contrast properties===

	* [[Lower triangular matrix]]: A matrix is upper triangular and lower triangular if and only if it is a [[diagonal matrix]].		* [[Lower triangular matrix]]: A matrix is upper triangular and lower triangular if and only if it is a [[diagonal matrix]].
	* [[Orthogonal matrix]]		* [[Orthogonal matrix]]
			* [[Symmetric matrix]]
			* [[Skew-symmetric matrix]]

Vipul: Created page with "{{square matrix property}} ==Definition== ===Verbal definition=== A square matrix is termed an '''upper triangular matrix''' if all its entries ''below'' the main dia..."

2014-05-01T17:54:13Z

Created page with "{{square matrix property}} ==Definition== ===Verbal definition=== A square matrix is termed an '''upper triangular matrix''' if all its entries ''below'' the main dia..."

New page

{{square matrix property}}

==Definition==

===Verbal definition===

A [[square matrix]] is termed an '''upper triangular matrix''' if all its entries ''below'' the [[main diagonal]] are zero.

===Algebraic description===

Suppose <math>n</math> is a positive integer. A <math>n \times n</math> [[square matrix]] <math>A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}</math> is termed an '''upper triangular matrix''' if the following holds:

<math>a_{ij} = \ \forall \ i > j \in \{ 1,2,\dots,n\}</math>

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::scalar matrix]] || scalar multiple of [[identity matrix]] || (via diagonal matrix) || (via diagonal matrix) || {{intermediate notions short|upper triangular matrix|scalar matrix}}
|-
| [[Weaker than::diagonal matrix]] || all entries not on the main diagonal are zero || || the matrix <math>\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\\end{pmatrix}</math> is a counterexample || {{intermediate notions short|upper triangular matrix|diagonal matrix}}
|-
| [[Weaker than::upper unitriangular matrix]] || upper triangular matrix with all diagonal entries equal to 1 || || || {{intermediate notions short|upper triangular matrix|upper unitriangular matrix}}
|-
| [[Weaker than::strictly upper triangular matrix]] || all entries on ''and'' below the main diagonal are zero || || || {{intermediate notions short|upper triangular matrix|strictly upper triangular matrix}}
|}
==Contrast properties===

* [[Lower triangular matrix]]: A matrix is upper triangular and lower triangular if and only if it is a [[diagonal matrix]].
* [[Orthogonal matrix]]