Diagonal matrix: Difference between revisions

Revision as of 02:58, 29 April 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

A diagonal matrix is a square matrix for which all the off-diagonal entries are zero, or equivalently, all nonzero entries are on the diagonal. Note that it is also possible that some (or even all) the diagonal entries are zero.

Encoding

A matrix that is known to be diagonal may simply be encoded using an ordered list of its diagonal entries. For a $n \times n$ matrix, this requires space for $n$ entries (in contrast with space for $n^{2}$ entries for an arbitrary square matrix).

Matrix operations

Unless otherwise specified, diagonal matrices are $n \times n$ diagonal matrices.

Matrix operation	Arithmetic complexity: number of additions	Arithmetic complexity: number of multiplications
Computation of the trace	$n - 1$	0
Computation of the determinant	0	$n - 1$
Addition of two such matrices	$n$	0
Multiplication of two such matrices	0	$n$
Multiplication by a $n \times p$ rectangular matrix (we allow $n = p$ )	0	$n p$

@@ Line 7: / Line 7: @@
 ==Encoding==
-{{fillin}}
+A matrix that is ''known'' to be diagonal may simply be encoded using an ordered list of its diagonal entries. For a <math>n \times n</math> matrix, this requires space for <math>n</math> entries (in contrast with space for <math>n^2</math> entries for an arbitrary square matrix).
+==Matrix operations==
+Unless otherwise specified, diagonal matrices are <math>n \times n</math> diagonal matrices.
+{| class="sortable" border="1"
+! Matrix operation !! Arithmetic complexity: number of additions !! Arithmetic complexity: number of multiplications
+|-
+| Computation of the [[trace]] || <math>n - 1</math> || 0
+|-
+| Computation of the [[determinant]] || 0 || <math>n - 1</math>
+|-
+| Addition of two such matrices || <math>n</math> || 0
+|-
+| Multiplication of two such matrices || 0 || <math>n</math>
+|-
+| Multiplication by a <math>n \times p</math> rectangular matrix (we allow <math>n = p</math>) || 0 || <math>np</math>
+|}