Diagonal matrix: Difference between revisions
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In [[matrix algebra]], a '''diagonal matrix''' is a [[square matrix]] for which only the entries on the main [[diagonal]] can be non-zero, and all the other, off-diagonal, entries are equal to zero. | In [[matrix algebra]], a '''diagonal matrix''' is a [[square matrix]] for which only the entries on the main [[diagonal]] can be non-zero, and all the other, off-diagonal, entries are equal to zero. | ||
The [[matrix addition|sum]] and [[matrix multiplication|product]] of diagonal matrices are again diagonal, and the diagonal matrices form a [[subring]] of the [[ring]] of square matrices: indeed for ''n''×''n'' matrices over a ring ''R'' this ring is [[ring isomorphism|isomorphic]] to the product ring ''R''<sup>''n''< | The [[matrix addition|sum]] and [[matrix multiplication|product]] of diagonal matrices are again diagonal, and the diagonal matrices form a [[subring]] of the [[ring]] of square matrices: indeed for ''n''×''n'' matrices over a ring ''R'' this ring is [[ring isomorphism|isomorphic]] to the product ring ''R''<sup>''n''</sup>. | ||
The [[zero matrix]] and the [[identity matrix]] are diagonal: they are the [[additive identity|additive]] and [[multiplicative identity]] respectively of the ring. | The [[zero matrix]] and the [[identity matrix]] are diagonal: they are the [[additive identity|additive]] and [[multiplicative identity]] respectively of the ring. | ||
A matrix over a field may be transformed into a diagonal matrix by a combination of [[row | A matrix over a field may be transformed into a diagonal matrix by a combination of [[row operation|row]] and [[column operation]]s to the the [[LDU decomposition]]. | ||
Revision as of 00:55, 10 December 2008
In matrix algebra, a diagonal matrix is a square matrix for which only the entries on the main diagonal can be non-zero, and all the other, off-diagonal, entries are equal to zero.
The sum and product of diagonal matrices are again diagonal, and the diagonal matrices form a subring of the ring of square matrices: indeed for n×n matrices over a ring R this ring is isomorphic to the product ring Rn.
The zero matrix and the identity matrix are diagonal: they are the additive and multiplicative identity respectively of the ring.
A matrix over a field may be transformed into a diagonal matrix by a combination of row and column operations to the the LDU decomposition.