Upper Triangular Matrix

In the mathematical discipline of linear algebra , a triangular matrix is ​​a special type of square matrix . A square matrix is ​​calledLower triangular ifall entriesabovethe main diagonalSimilarly, a square matrix is ​​calledUpper triangular ifall entriesbelowthe main diagonal Because matrix equations containing triangular matrices are easy to solve, they are very important in numerical analysis . By the LU decomposition algorithm, an inverse matrix can be written as the product of a lower triangular matrix L and an upper triangular matrix U if and only if all its principal minors are non-zero.

Description

a matrix of the form

{\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 bottom triangular matrix or left triangular matrix , and a matrix of uniform form

{\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}}}

called the upper triangular matrix or right triangular matrix . A lower or left triangular matrix is ​​usually represented with the variable l , and an upper or right triangular matrix is ​​usually represented with the variable u or r .

A matrix that is both an upper and a lower triangular is diagonal . Matrices that are similar to triangular matrices are said to be triangularisable .

A non-square (or sometimes none) matrix with zeros above (below) the diagonal is called a lower (upper) trapezoidal matrix. The non-zero entries form the shape of a trapezoid .

Example

this matrix

{\displaystyle {\begin{bmatrix}1&4&1\\0&6&4\\0&0&1\\\end{bmatrix}}}

is upper triangular and this matrix

{\displaystyle {\begin{bmatrix}1&0&0\\2&8&0\\4&9&7\\\end{bmatrix}}}

The bottom is triangular.

Front and back replacement

It is very easy to solve a matrix equation in the form or by an iterative process called forward substitution for the lower triangular matrix and similarly backward substitution for the upper triangular matrix . The procedure is so called because for the lower triangular matrix, one calculates first , then substituting that forward into the next equation to solve for , and to . repeats through . In an upper triangular matrix, one works backwards , first computing , then substituting it back into the previous equation to solve

{\displaystyle L\mathbf {x} =\mathbf {b} }{\displaystyle U\mathbf {x} =\mathbf {b} }x_{1}x_{2}x_{n}x_{n}x_{n-1}

and repeat through .x1 Note that this does not require inverting the matrix.

Forward replacement

The matrix equation x = b can be written as the system of linear equations

{\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}}}

Note that the first equation ( ) involves only , and thus no . can solve for directly. The second equation contains only and , and thus can be solved once for a substitute in the already solved value . Continuing thus, the -th equation only includes , and one can solve for using the previously solved values ​​for .

\ell _{1,1}x_{1}=b_{1}x_{1}x_{1}x_{1}x_{2}x_{1}kx_{1},\dots ,x_{k}x_{k}x_{1},\dots ,x_{k-1}
{\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 a similar way, only working backwards.

Application

Forward substitution is used in financial bootstrapping to create a yield curve .

Virtue

Transpose an upper triangular matrix to a lower triangular matrix and vice versa. A matrix that is both symmetric and triangular is diagonal. In a similar vein, a matrix that is both in general  (meaning a = aa * , where * is the conjugate transpose ) and a triangular is also diagonal. This can be seen by looking at the diagonal entries of a and AA * .

The determinant and permanent of a triangular matrix are equal to the product of the diagonal entries, as can be checked by direct computation.

is actually more true: the eigenvalues ​​of a triangular matrix are exactly its diagonal entries. Furthermore, each eigenvalue occurs exactly k times on the diagonal, where k is its algebraic multiplicity , i.e., it has as the root of the characteristic polynomial

{\displaystyle p_{A}(x)=\det(xI-A)}K A. 

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

{\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 properties). See this to see that the triangle is even and therefore its determinant is the product of its diagonal entries .

xI-A{\displaystyle \det(xI-A)}{\displaystyle (x-a_{11})(x-a_{22})\cdots (x-a_{nn})}

Especially

Untriangular matrix

If the entries on the principal diagonal of a (upper or lower) triangular matrix are all 1, then the matrix is ​​said to be (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 a unit matrix , and the notion of a normed triangular matrix has nothing to do with the ideal of the matrix .

All unitriangular matrices are unipotent.

Strictly triangular matrix

If all entries on the principal diagonal of a (upper or lower) triangular matrix are 0, then the matrix is ​​said to be strictly (upper or lower) triangular .

All strictly triangular matrices are nilpotent .

Atomic Triangular Matrix

An atomic (upper or lower) triangular matrix

is ​​a special form of the unitringular matrix where, except for the entries in a column, all off-diagonal elements are zero . Such a matrix is ​​also called a Frobenius matrix , a Gauss matrix or a Gauss transformation matrix .

Tri-aptitude

A matrix which is similar to a triangular matrix is ​​called a triangular matrix . In short, this is equivalent to a flag constant: the upper triangular matrices are precisely those that preserve the standard flag , given by the standard ordered basis and the resulting flag . All flags are conjugate (as opposed to the usual linear group of bases). acts transitively), so any matrix that stabilizes a flag is the same as one that stabilizes a standard flag.

(e_{1},\ldots ,e_{n})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}.

Any complex square matrix is ​​triangular. [1] In fact, a matrix containing all of the eigenvalues ​​of more than one field ( for example, any matrix over a closed field of algebra ) is similar to a triangular matrix. This can be proved using induction on the fact that A has an eigenvector, by taking the quotient space by the eigenvector and showing that A stabilizes a flag, and thus with respect to the base of that flag. is triangular.

A more precise statement is given by the Jordan normal form theorem, which states that in this case, A is identical to an upper triangular matrix of a very special form. The simple triangulation result is often sufficient, and is used in proving Jordan’s normal form theorem in any case. [1] [2]

In the case of complex matrices, it is possible to say more about triangulation, that is, any square matrix A has a Schur decomposition. This means that A is unitarily equivalent to an upper triangular matrix (i.e. identical, using the unitary matrix as a transformation of the basis); It carries a Hermitian base for the flag.

Simultaneous tri-qualification

A set of matrices is called A1,……AkTriangularizable together if there is a base under which all those vertices are triangular; Equivalently, if theyare upper triangular by asingleP. Such a set of matrices can be more easily understood by considering the algebra of the matrices that arise, i.e. all polynomialtargetisomorphic trigonometry means that this algebra is conjugate to the Lie subalgebra of the upper triangular matrix, and this algebra is equivalent to which is the Lie subalgebra of the Borel subalgebra.

A_{i},K[A_{1},\ldots ,A_{k}].

The basic result is that (on an algebraically closed field), the incoming matrices are, or more generally, simultaneous triangulation. This can be proved by first showing that the incoming matrix has a common eigenvector, and then joining on the same dimension as before. This was proved by Frobenius, who began in 1878 for a commuting pair, as discussed in Commuting Matrices. For single matrices, these can be triangulated by a unitary matrix over the complex numbers.

A,BA_{1},\ldots ,A_{k}

The fact that the incoming matrix has a normal eigenvector can be interpreted as a consequence of Hilbert’s Nullstellensatz: the coming matrix forms a commutative algebra above which can be interpreted as a variation in k -dimensional affine space. , and the existence of a (normal) eigenvalue (and hence a normal eigenvector) corresponds to this variation containing a point (non-empty), which is the material (weak) Nullstellensatz. In algebraic terms, these operators correspond to algebraic representations of polynomial algebras in k variables.

K[A_{1},\ldots ,A_{k}]K[x_{1},\ldots ,x_{k}]

This is generalized by Lie’s theorem, which shows that any representation of a solvable Lie algebra is simultaneously upper triangular, the case of matrices being the abelian Lie algebra case, the abelian a Fortiori solvable.

More generally and precisely, a set of matrices is triangulated together if and only if the matrix is ​​nilpotent for all polynomials p in k non -variable-commuting, where is the commutator; For commutation the commutator disappears so it holds. This was proved (Drazin, Dange and Gruenberg 1951); A brief proof is given in (Prasolov 1994, pp. 178-179). One direction is clear: if the matrices are simultaneously triangular, then is is strictly upper triangularizable (hence nilpotent), protected by multiplication by any which —or a combination thereof—will still have 0s on the diagonal at the base of the triangle.

A_{1},\ldots ,A_{k}p(A_{1},\ldots ,A_{k})[A_{i},A_{j}] [A_{i},A_{j}]A_{i}[A_{i},A_{j}]A_{k}

Algebra of triangular arrays

Upper Triangular Matrix
Upper Triangular Matrix

Upper triangularity is preserved by several functions:

  • The sum of two upper triangular matrices is the upper triangular one.
  • The product of two upper triangular matrices is the upper triangular one.
  • The inverse of an upper triangular matrix, where it exists, is an upper triangular one.
  • The product of an upper triangular matrix and a scalar is an upper triangular one.

Together these facts mean that the upper triangular matrices form a sub-algebra of the associative algebra of square matrices for a given shape. In addition, it also shows that the upper triangular matrix can be viewed as a Lie sub-algebra of the Lie algebra of a square matrix of a fixed size, where the Lie bracket given by the commutator ab – ba [ a , b . ] is . The Lie algebra of all upper triangular matrices is a solvable Lie algebra. It is often referred to as the Borel sub-algebra of the Lie algebra of all square matrices.

All these results hold if the upper triangular is replaced by the lower triangular ; In particular the lower triangular matrices also form a Lie algebra. However, operations that combine upper and lower triangular matrices do not in general produce triangular matrices. For example, the sum of the upper and lower triangular matrices can be any matrix; The product of the lower triangle with the upper triangular matrix is ​​not necessarily triangular either.

The set of non-triangular matrices forms a false group.

The set of strictly upper (or lower) triangular matrices forms a nilpotent Lie algebra, denoted as the derivative of this algebra is the Lie algebra , the Lie algebra of all upper triangular matrices; In symbols, there is, at the same time, the Lie algebra of the Lie group of non-triangular matrices.

{\mathfrak {n}}.{\mathfrak {b}}{\mathfrak {n}}=[{\mathfrak {b}},{\mathfrak {b}}].{\mathfrak {n}}

In fact, according to Engel’s theorem, any finite-dimensional nilpotent Lie algebra is strictly conjugate with a sub-algebra of an upper triangular matrix, that is, a finite-dimensional nilpotent Lie algebra is simultaneously strictly upper triangular.

Functional analysis in the algebra of upper triangular matrices has a natural generalization that generates nested algebras on Hilbert spaces.

Borel Subgroups and Borel Subalgebras

The set of inverse triangular matrices of a given type (upper or lower) forms a group, in fact a Lie group, which is a subset of the general linear group of all inverse matrices. A triangular matrix is ​​invertible exactly when its diagonal entries are invertible (non-zero).

On real numbers, this group is truncated according to the components each diagonal entry is positive or negative. The identity component is an inverse triangular matrix with positive entries on the diagonal, and the group of all inverse triangular matrices is a semidirect product of this group and the group of diagonal matrices on the diagonal, with corresponding components.2^{n}\pm 1

Lie algebra The inverse upper triangular matrix of the Lie group is the set of all upper triangular matrices, not necessarily the inverse, and is an interpretable Lie algebra. These are, respectively, the standard Borel subgroup B and the standard Borel subalgebra of the Lie group gl n . {\mathfrak {b}}Lie algebra of gl n .

The upper triangular matrices are precisely those that stabilize the standard flag. The inverses of them form a subset of the general linear group, whose conjugate subgroups are those that are defined as stabilizers of some (other) absolute flag. These subgroups are the Borel subgroups. The group of invertible lower triangular matrices is one such subgroup, as it is the stabilizer of the standard flag attached to the standard base in reverse order.

The stabilizer of a partial flag, obtained by forgetting parts of the standard flag, can be described as a set of block upper triangular matrices (but its elements are not all triangular matrices ). There are conjugate subgroups of such groups which are defined as stabilizers of some partial flag. These subgroups are called parabolic subgroups.

Example

The 2 × 2 upper unitriangular matrix of the group is isomorphic to the additive group of the area of ​​the scalars; In the case of complex numbers, this corresponds to a group made up of parabolic Möbius transformations; The 3×3 upper triangular matrix forms the Heisenberg group.