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''</sup>. | The [[matrix addition|sum]] and [[matrix multiplication|product]] of diagonal matrices are again diagonal, and the diagonal matrices form a [[subring]] of the [[ring (mathematics)|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>. | ||
==Examples== | |||
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. | ||
==Properties== | |||
The diagonal entries are the [[eigenvalue]]s of a diagonal matrix. | |||
The [[determinant]] of a diagonal matrix is the product of the diagonal elements. | The [[determinant]] of a diagonal matrix is the product of the diagonal elements. | ||
A matrix over a field may be transformed into a diagonal matrix by a combination of [[row operation|row]] and [[column operation]]s: this is the [[LDU decomposition]]. | A matrix over a field may be transformed into a diagonal matrix by a combination of [[row operation|row]] and [[column operation]]s: this is the [[LDU decomposition]]. | ||
==Diagonalizable matrix== | |||
A '''diagonalizable''' matrix is a [[square matrix]] which is [[similar matrices|similar]] to a diagonal matrix: that is, ''A'' is diagonalizable if there exists an [[invertible matrix]] ''P'' such that <math>P^{-1}AP</math> is diagonal. The following conditions are equivalent: | |||
* ''A'' is diagonalizable; | |||
* The [[minimal polynomial]] of ''A'' has no repeated roots; | |||
* ''A'' is ''n''×''n'' and has ''n'' [[linear independence|linearly independent]] [[eigenvector]]s.[[Category:Suggestion Bot Tag]] | |||
Latest revision as of 16:01, 6 August 2024
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.
Examples
The zero matrix and the identity matrix are diagonal: they are the additive and multiplicative identity respectively of the ring.
Properties
The diagonal entries are the eigenvalues of a diagonal matrix.
The determinant of a diagonal matrix is the product of the diagonal elements.
A matrix over a field may be transformed into a diagonal matrix by a combination of row and column operations: this is the LDU decomposition.
Diagonalizable matrix
A diagonalizable matrix is a square matrix which is similar to a diagonal matrix: that is, A is diagonalizable if there exists an invertible matrix P such that is diagonal. The following conditions are equivalent:
- A is diagonalizable;
- The minimal polynomial of A has no repeated roots;
- A is n×n and has n linearly independent eigenvectors.