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Text of the page (random words):
quare matrix a square matrix is called lower triangular if all the entries above the main diagonal are zero similarly a square matrix is called upper triangular if all the entries below the main diagonal are zero because matrix equations with triangular matrices are easier to solve they are very important in numerical analysis by the lu decomposition algorithm an invertible matrix may be written as the product of a lower triangular matrix l and an upper triangular matrix u if and only if all its leading principal minors are non zero contents 1 description 1 1 examples 2 forward and back substitution 2 1 forward substitution 2 2 applications 3 properties 4 special forms 4 1 unitriangular matrix 4 2 strictly triangular matrix 4 3 atomic triangular matrix 5 triangularisability 5 1 simultaneous triangularisability 6 algebras of triangular matrices 6 1 borel subgroups and borel subalgebras 6 2 examples 7 see also 8 references description edit a matrix of the form l ℓ 1 1 0 ℓ 2 1 ℓ 2 2 ℓ 3 1 ℓ 3 2 ℓ n 1 ℓ n 2 ℓ n n 1 ℓ n n displaystyle l begin bmatrix ell _ 1 1 0 ell _ 2 1 ell _ 2 2 ell _ 3 1 ell _ 3 2 ddots vdots vdots ddots ddots ell _ n 1 ell _ n 2 ldots ell _ n n 1 ell _ n n end bmatrix is called a lower triangular matrix or left triangular matrix and analogously a matrix of the form u u 1 1 u 1 2 u 1 3 u 1 n u 2 2 u 2 3 u 2 n u n 1 n 0 u n n displaystyle u begin bmatrix u_ 1 1 u_ 1 2 u_ 1 3 ldots u_ 1 n u_ 2 2 u_ 2 3 ldots u_ 2 n ddots ddots vdots ddots u_ n 1 n 0 u_ n n end bmatrix is called an upper triangular matrix or right triangular matrix a lower or left triangular matrix is commonly denoted with the variable l and an upper or right triangular matrix is commonly denoted with the variable u or r a matrix that is both upper and lower triangular is diagonal matrices that are similar to triangular matrices are called triangularisable a non square or sometimes any matrix with zeros above below the diagonal is called a lower upper trapezoidal matrix the non zero entries form the shape of a trapezoid examples edit this matrix 1 4 1 0 6 4 0 0 1 displaystyle begin bmatrix 1 4 1 0 6 4 0 0 1 end bmatrix is upper triangular and this matrix 1 0 0 2 96 0 4 9 69 displaystyle begin bmatrix 1 0 0 2 96 0 4 9 69 end bmatrix is lower triangular forward and back substitution edit a matrix equation in the form l x b displaystyle l mathbf x mathbf b or u x b displaystyle u mathbf x mathbf b is very easy to solve by an iterative process called forward substitution for lower triangular matrices and analogously back substitution for upper triangular matrices the process is so called because for lower triangular matrices one first computes x 1 displaystyle x_ 1 then substitutes that forward into the next equation to solve for x 2 displaystyle x_ 2 and repeats through to x n displaystyle x_ n in an upper triangular matrix one works backwards first computing x n displaystyle x_ n then substituting that back into the previous equation to solve for x n 1 displaystyle x_ n 1 and repeating through x 1 displaystyle x_ 1 notice that this does not require inverting the matrix forward substitution edit the matrix equation l x b can be written as a system of linear equations ℓ 1 1 x 1 b 1 ℓ 2 1 x 1 ℓ 2 2 x 2 b 2 ℓ m 1 x 1 ℓ m 2 x 2 ℓ m m x m b m displaystyle begin matrix ell _ 1 1 x_ 1 b_ 1 ell _ 2 1 x_ 1 ell _ 2 2 x_ 2 b_ 2 vdots vdots ddots vdots ell _ m 1 x_ 1 ell _ m 2 x_ 2 dotsb ell _ m m x_ m b_ m end matrix observe that the first equation ℓ 1 1 x 1 b 1 displaystyle ell _ 1 1 x_ 1 b_ 1 only involves x 1 displaystyle x_ 1 and thus one can solve for x 1 displaystyle x_ 1 directly the second equation only involves x 1 displaystyle x_ 1 and x 2 displaystyle x_ 2 and thus can be solved once one substitutes in the already solved value for x 1 displaystyle x_ 1 continuing in this way the k displaystyle k th equation only involves x 1 x k displaystyle x_ 1 dots x_ k and one can solve for x k displaystyle x_ k using the previously solved values for x 1 x k 1 displaystyle x_ 1 dots x_ k 1 the resulting formulas are x 1 b 1 ℓ 1 1 x 2 b 2 ℓ 2 1 x 1 ℓ 2 2 x m b m i 1 m 1 ℓ m i x i ℓ m m displaystyle begin aligned x_ 1 frac b_ 1 ell _ 1 1 x_ 2 frac b_ 2 ell _ 2 1 x_ 1 ell _ 2 2 vdots x_ m frac b_ m sum _ i 1 m 1 ell _ m i x_ i ell _ m m end aligned a matrix equation with an upper triangular matrix u can be solved in an analogous way only working backwards applications edit forward substitution is used in financial bootstrapping to construct a yield curve properties edit the transpose of an upper triangular matrix is a lower triangular matrix and vice versa a matrix which is both symmetric and triangular is diagonal in a similar vein a matrix which is both normal meaning a a aa where a is the conjugate transpose and triangular is also diagonal this can be seen by looking at the diagonal entries of a a and aa the determinant and 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 eigenvalues 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 as a root of the characteristic polynomial p a x det x i a displaystyle p_ a x det xi a of a in other words the characteristic polynomial of a triangular n n matrix a is exactly p a x x a 11 x a 22 x a n n displaystyle p_ a x x a_ 11 x a_ 22 cdots x a_ nn that is the unique degree n polynomial whose roots are the diagonal entries of a with multiplicities to see this observe that x i a displaystyle xi a is also triangular and hence its determinant det x i a displaystyle det xi a is the product of its diagonal entries x a 11 x a 22 x a n n displaystyle x a_ 11 x a_ 22 cdots x a_ nn 1 special forms edit unitriangular matrix edit if the entries on the main diagonal of a upper or lower triangular matrix are all 1 the matrix is called upper or lower unitriangular other names used for these matrices are unit upper or lower triangular or very rarely normed upper or lower triangular however a unit triangular matrix is not the same as the unit matrix and a normed triangular matrix has nothing to do with the notion of matrix norm all finite unitriangular matrices are unipotent strictly triangular matrix edit if all of the entries on the main diagonal of a upper or lower triangular matrix are also 0 the matrix is called strictly upper or lower triangular all finite strictly triangular matrices are nilpotent of index at most n as a consequence of the cayley hamilton theorem atomic triangular matrix edit main article frobenius matrix an atomic upper or lower triangular matrix is a special form of unitriangular matrix where all of the off diagonal elements 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 edit a matrix that is similar to a triangular matrix is referred to as triangularizable abstractly this is equivalent to stabilizing a flag upper triangular matrices are precisely those that preserve the standard flag which is given by the standard ordered basis e 1 e n displaystyle e_ 1 ldots e_ n and the resulting flag 0 e 1 e 1 e 2 e 1 e n k n displaystyle 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 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 stabilizes the standard flag any complex square matrix is triangularizable 1 in fact a matrix a over a 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 stabilizes 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 1 2 in the case of complex matrices it is possible to say more about triangularization namely that any square matrix a has a schur decomposition this means that a is unitarily equivalent i e similar using a unitary matrix as change of basis to an upper triangular matrix this follows by taking an hermitian basis for the flag simultaneous triangularisability edit see also simultaneously diagonalizable a set of matrices a 1 a k displaystyle a_ 1 ldots a_ k are said to be simultaneously triangularisable if there is a basis under which they are all upper triangular equivalently if they are upper triangularizable by a single similarity matrix p such a set of matrices is more easily understood by considering the algebra of matrices it generates namely all polynomials in the a i displaystyle a_ i denoted k a 1 a k displaystyle k a_ 1 ldots a_ k simultaneous triangularizability means that this algebra is conjugate into the lie subalgebra of upper triangular matrices and is equivalent to this algebra being a lie subalgebra of a borel subalgebra the basic result is that over an algebraically closed field the commuting matrices a b displaystyle a b or more generally a 1 a k displaystyle a_ 1 ldots a_ k are simultaneously triangularizable this can be proven by first showing that commuting matrices have a common eigenvector and then inducting on dimension as before this was proven by frobenius starting in 1878 for a commuting pair as discussed at commuting matrices as for a single matrix over the complex numbers these can be triangularized by unitary matrices the fact that commuting matrices have a common eigenvector can be interpreted as a result of hilbert s nullstellensatz commuting matrices form a commutative algebra k a 1 a k displaystyle k a_ 1 ldots a_ k over k x 1 x k displaystyle k x_ 1 ldots x_ k which can be interpreted as a variety in k dimensional affine space and the existence of a common eigenvalue and hence a common eigenvector corresponds to this variety having a point being non empty which is the content of the weak nullstellensatz citation needed in algebraic terms these operators correspond to an algebra representation of the polynomial algebra in k variables this is generalized by lie s theorem which shows that any representation of a solvable lie algebra is simultaneously upper triangularizable the case of commuting matrices being the abelian lie algebra case abelian being a fortiori solvable more generally and precisely a set of matrices a 1 a k displaystyle a_ 1 ldots a_ k is simultaneously triangularisable if and only if the matrix p a 1 a k a i a j displaystyle p a_ 1 ldots a_ k a_ i a_ j is nilpotent for all polynomials p in k non commuting variables where a i a j displaystyle a_ i a_ j is the commutator for commuting a i displaystyle a_ i the commutator vanishes so this holds this was proven by drazin dungey and gruenberg in 1951 3 a brief proof is given by prasolov in 1994 4 one direction is clear if the matrices are simultaneously triangularisable then a i a j displaystyle a_ i a_ j is strictly upper triangularizable hence nilpotent which is preserved by multiplication by any a k displaystyle a_ k or combination thereof it will still have 0s on the diagonal in the triangularizing basis algebras of triangular matrices edit binary lower unitriangular toeplitz matrices multiplied using f 2 operations they form the cayley table of z 4 and correspond to powers of the 4 bit gray code permutation upper triangularity is preserved by many operations the sum of two upper triangular matrices is upper triangular the product of two upper triangular matrices is upper triangular the inverse of an upper triangular matrix if it exists is upper triangular the product of an upper triangular matrix and a scalar is upper triangular together these facts mean that the upper triangular matrices form a subalgebra of the associative algebra of square matrices for a given size additionally this also shows that the upper triangular matrices can be viewed as a lie subalgebra of the lie algebra of square matrices of a fixed size where the lie bracket a b given by the commutator ab ba the lie algebra of all upper triangular matrices is a solvable lie algebra it is often referred to as a borel subalgebra of the lie algebra of all square matrices all these results hold if upper triangular is replaced by lower triangular throughout in particular the lower triangular matrices also form a lie algebra however operations mixing upper and lower triangular matrices do not in general produce triangular matrices for instance the sum of an upper and a lower triangular matrix can be any matrix the product of a lower triangular with an upper triangular matrix is not necessarily triangular either the set of unitriangular matrices forms a lie group the set of strictly upper or lower triangular matrices forms a nilpotent lie algebra denoted n displaystyle mathfrak n this algebra is the derived lie algebra of b displaystyle mathfrak b the lie algebra of all upper triangular matrices in symbols n b b displaystyle mathfrak n mathfrak b mathfrak b in addition n displaystyle mathfrak n is the lie algebra of the lie group of unitriangular matrices in fact by engel s theorem any finite dimensional nilpotent lie algebra is conjugate to a subalgebra of the strictly upper triangular matrices that is to say a finite dimensional nilpotent lie algebra is simultaneously strictly upper triangularizable algebras of upper triangular matrices have a natural generalization in functional analysis which yields nest algebras on hilbert spaces see also affine group borel subgroups and borel subalgebras edit main articles borel subgroup and borel subalgebra the set of invertible triangular matrices of a given kind upper or lower forms a group indeed a lie group which is a subgroup of the general linear group of all invertible matrices a triangular matrix is invertible precisely when its diagonal entries are invertible non zero over the real numbers this group is disconnected having 2 n displaystyle 2 n components accordingly as each diagonal entry is positive or negative the identity component is invertible triangular matrices with positive entries on the diagonal and the group of all invertible triangular matrices is a semidirect product of this group and the group of diagonal matrices with 1 displaystyle pm 1 on the diagonal corresponding to the components the lie algebra of the lie group of invertible upper triangular matrices is the set of all upper triangular matrices not necessarily invertible and is a solvable lie algebra these are respectively the standard borel subgroup b o...