Upper triangular matrix: Difference between revisions
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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}} | ||
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==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]] |
Revision as of 17:54, 1 May 2014
This article defines a property that can be evaluated for a square matrix. In other words, given a square matrix (a matrix with an equal number of rows and columns) the matrix either satisfies or does not satisfy the property.
View other properties of square matrices
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 is a positive integer. A square matrix is termed an upper triangular matrix if the following holds:
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
scalar matrix | scalar multiple of identity matrix | (via diagonal matrix) | (via diagonal matrix) | Diagonal matrix|FULL LIST, MORE INFO |
diagonal matrix | all entries not on the main diagonal are zero | the matrix Failed to parse (unknown function "\begin{pmatrix}"): {\displaystyle \begin{pmatrix} 1 & 1 \\ 0 & 1 \\\\end{pmatrix}} is a counterexample | |FULL LIST, MORE INFO | |
upper unitriangular matrix | upper triangular matrix with all diagonal entries equal to 1 | |FULL LIST, MORE INFO | ||
strictly upper triangular matrix | all entries on and below the main diagonal are zero | |FULL LIST, MORE INFO |
Contrast properties
- Lower triangular matrix: A matrix is upper triangular and lower triangular if and only if it is a diagonal matrix.
- Orthogonal matrix
- Symmetric matrix
- Skew-symmetric matrix