Triangular matrix: Difference between revisions

Revision as of 10:20, 2 March 2021

Revision as of 10:08, 9 March 2021

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\end{align}</math>

A matrix equation with an upper triangular matrix '''U''' can be solved in an analogous way, only working backwards.

===Applications===

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The [[determinant]] and [[Permanent (mathematics)|permanent]] of a triangular matrix equal the product of the diagonal entries, as can be checked by direct computation.

In fact more is true: the [[eigenvalue]]s of a triangular matrix are exactly its diagonal entries. Moreover, each eigenvalue occurs exactly ''k'' times on the diagonal, where ''k'' is its [[algebraic multiplicity]], that is, its [[Multiplicity of a root of a polynomial|multiplicity as a root]] of the [[characteristic polynomial]] <math>p_A(x)=\det(xI-A)</math> of ''A''. In other words, the characteristic polynomial of a triangular ''n''×''n'' matrix ''A'' is exactly

Moreover, each eigenvalue occurs exactly ''k'' times on the diagonal, where ''k'' is its [[algebraic multiplicity]], that is, its [[Multiplicity of a root of a polynomial|multiplicity as a root]] of the [[characteristic polynomial]] <math>p_A(x)=\operatorname{det}(xI-A)</math> of ''A''.

In other words, the characteristic polynomial of a triangular ''n''×''n'' matrix ''A'' is exactly

: <math>p_A(x) = (x-a_{11})(x-a_{22})\cdots(x-a_{nn})</math>,

that is, the unique degree ''n'' polynomial whose roots are the diagonal entries of ''A'' (with multiplicities).

To see this, observe that <math>xI-A</math> is also triangular and hence its determinant <math>\operatorname{det}(xI-A)</math> is the product of its diagonal entries <math>(x-a_{11})(x-a_{22})\cdots(x-a_{nn})</math>.<ref name="axler">{{Harv|Axler|1996|loc=pp. 86&ndash;87, 169}}</ref>

==Special forms==

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An '''atomic''' (upper or lower) '''triangular matrix''' is a special form of unitriangular matrix, where all of the [[off-diagonal element]]s are zero, except for the entries in a single column. Such a matrix is also called a '''Frobenius matrix''', a '''Gauss matrix''', or a '''Gauss transformation matrix'''.

==Triangularisability{{Anchor|Triangularizability}}==

A matrix that is [[similar matrix|similar]] to a triangular matrix is referred to as '''triangularizable'''. Abstractly, this is equivalent to stabilizing a [[flag (linear algebra)|flag]]: upper triangular matrices are precisely those that preserve the [[standard flag]], which is given by the standard ordered basis <math>(e_1,\ldots,e_n)</math> and the resulting flag <math>0 < \left\langle e_1\right\rangle < \left\langle e_1,e_2\right\rangle < \cdots < \left\langle e_1,\ldots,e_n \right\rangle = K^n.</math> All flags are conjugate (as the general linear group acts transitively on bases), so any matrix that stabilises a flag is similar to one that stabilisesstabilizes the standard flag.

Any complex square matrix is triangularizable.<ref name="axler"/> In fact, a matrix ''A'' over a [[field (mathematics)|field]] containing all of the eigenvalues of ''A'' (for example, any matrix over an [[algebraically closed field]]) is similar to a triangular matrix. This can be proven by using induction on the fact that ''A'' has an eigenvector, by taking the quotient space by the eigenvector and inducting to show that ''A'' stabilisesstabilizes a flag, and is thus triangularizable with respect to a basis for that flag.

A more precise statement is given by the [[Jordan normal form]] theorem, which states that in this situation, ''A'' is similar to an upper triangular matrix of a very particular form. The simpler triangularization result is often sufficient however, and in any case used in proving the Jordan normal form theorem.<ref name="axler"/><ref name="herstein">{{Harv|Herstein|1975|loc=pp. 285&ndash;290}}</ref>

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=== Examples ===

The group of 2 by 22×2 upper unitriangular matrices is [[isomorphic]] to the [[Abelian group|additive group]] of the field of scalars; in the case of complex numbers it corresponds to a group formed of parabolic [[Möbius transformation]]s; the 3 by 33×3 upper unitriangular matrices form the [[Heisenberg group]].

== See also ==