Bandwidth of a matrix - Revision history

Vipul at 17:40, 1 May 2014

2014-05-01T17:40:19Z

← Older revision		Revision as of 17:40, 1 May 2014
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	When we say that a matrix has a given bandwidth (respectively, a given left half-bandwidth or a given right half-bandwidth) what we mean is that the bandwidth (respectively, the left or right half-bandwidth) is ''at most'' that quantity, not that it is necessarily exactly equal to that quantity.		When we say that a matrix has a given bandwidth (respectively, a given left half-bandwidth or a given right half-bandwidth) what we mean is that the bandwidth (respectively, the left or right half-bandwidth) is ''at most'' that quantity, not that it is necessarily exactly equal to that quantity.

			===The notion of band matrix or banded matrix===

			The term '''band matrix''' or '''banded matrix''' is used for a matrix whose bandwidth is reasonably small.

	==Particular cases==		==Particular cases==

Vipul at 17:16, 1 May 2014

2014-05-01T17:16:39Z

← Older revision		Revision as of 17:16, 1 May 2014
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	Suppose <math>n</math> is a positive integer and <math>A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}</math> is a <math>n \times n</math> [[square matrix]].		Suppose <math>n</math> is a positive integer and <math>A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}</math> is a <math>n \times n</math> [[square matrix]].

	The '''left half-bandwidth''' of <math>A<math> is defined as the smallest positive integer <math>k_1</math> such that <math>a_{ij} = 0</math> whenever <math>j < i - k_1</math>. In other words, entries that are more than <math>k_1</math> positions below the [[main diagonal]] are zero.		The '''left half-bandwidth''' of <math>A</math> is defined as the smallest positive integer <math>k_1</math> such that <math>a_{ij} = 0</math> whenever <math>j < i - k_1</math>. In other words, entries that are more than <math>k_1</math> positions below the [[main diagonal]] are zero.

	The '''right half-bandwidth''' of <math>A</math> is defined as the smallest positive integer <math>k_2</math> such that <math>a_{ij} = 0</math> whenever <math>j < i + k_2</math>. In other words, entries that are more than <math>k_2</math> positions above the [[main diagonal]] are zero.		The '''right half-bandwidth''' of <math>A</math> is defined as the smallest positive integer <math>k_2</math> such that <math>a_{ij} = 0</math> whenever <math>j < i + k_2</math>. In other words, entries that are more than <math>k_2</math> positions above the [[main diagonal]] are zero.

Vipul at 17:15, 1 May 2014

2014-05-01T17:15:54Z

← Older revision		Revision as of 17:15, 1 May 2014
Line 12:		Line 12:

	When we say that a matrix has a given bandwidth (respectively, a given left half-bandwidth or a given right half-bandwidth) what we mean is that the bandwidth (respectively, the left or right half-bandwidth) is ''at most'' that quantity, not that it is necessarily exactly equal to that quantity.		When we say that a matrix has a given bandwidth (respectively, a given left half-bandwidth or a given right half-bandwidth) what we mean is that the bandwidth (respectively, the left or right half-bandwidth) is ''at most'' that quantity, not that it is necessarily exactly equal to that quantity.

			==Particular cases==

			{\| class="sortable" border="1"
			! Matrix type (all matrices are <math>n \times n</math>) !! Left half-bandwidth (upper bound) !! Right half-bandwidth (upper bound) !! Bandwidth (upper bound)
			\|-
			\| [[diagonal matrix]] \|\| 0 \|\| 0 \|\| 1
			\|-
			\| [[tridiagonal matrix]] \|\| 1 \|\| 1 \|\| 3
			\|-
			\| [[pentadiagonal matrix]] \|\| 2 \|\| 2 \|\| 5
			\|-
			\| [[upper triangular matrix]] \|\| 0 \|\| <math>n - 1</math> \|\| <math>n</math>
			\|-
			\| [[lower triangular matrix]] \|\| <math>n - 1</math> \|\| 0 \|\| <math>n</math>
			\|}

Vipul: Created page with "==Definition== Suppose $n$ is a positive integer and $A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}$ is a $n \times n$ square matrix. The..."

2014-05-01T17:12:02Z

Created page with "==Definition== Suppose <math>n</math> is a positive integer and <math>A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}</math> is a <math>n \times n</math> square matrix. The..."

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==Definition==

Suppose <math>n</math> is a positive integer and <math>A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}</math> is a <math>n \times n</math> [[square matrix]].

The '''left half-bandwidth''' of <math>A<math> is defined as the smallest positive integer <math>k_1</math> such that <math>a_{ij} = 0</math> whenever <math>j < i - k_1</math>. In other words, entries that are more than <math>k_1</math> positions below the [[main diagonal]] are zero.

The '''right half-bandwidth''' of <math>A</math> is defined as the smallest positive integer <math>k_2</math> such that <math>a_{ij} = 0</math> whenever <math>j < i + k_2</math>. In other words, entries that are more than <math>k_2</math> positions above the [[main diagonal]] are zero.

The '''banwidth''' is defined as <math>k_1 + k_2 + 1</math>, where <math>k_1,k_2</math> are the left and right half-bandwidths respectively.

===Ambiguity with terminology===

When we say that a matrix has a given bandwidth (respectively, a given left half-bandwidth or a given right half-bandwidth) what we mean is that the bandwidth (respectively, the left or right half-bandwidth) is ''at most'' that quantity, not that it is necessarily exactly equal to that quantity.

Bandwidth of a matrix - Revision history

Vipul at 17:40, 1 May 2014

Vipul at 17:16, 1 May 2014

Vipul at 17:15, 1 May 2014

Vipul: Created page with "==Definition== Suppose n is a positive integer and A = (a_{ij})_{1 \le i \le n, 1 \le j \le n} is a n \times n square matrix. The..."

Vipul: Created page with "==Definition== Suppose $n$ is a positive integer and $A = (a_{ij})_{1 \le i \le n, 1 \le j \le n}$ is a $n \times n$ square matrix. The..."