Triangle matrix
In linear algebra, a triangular matrix is a special type of square matrix whose elements above or below its main diagonal or minor diagonal are zero. Because systems of linear equations with triangular matrices are much easier to solve, triangular matrices are used in numerical analysis to solve systems of linear equations, compute inverses, and determinants of matrices. The LU decomposition method allows decomposing any invertible matrix as a product of a lower triangular matrix L and an upper triangular matrix U.
Description
A square matrix of order n is said to be upper triangular if it is of the form:
- U=[chuckles]u11u12u13...u1n0u22u23...u2n00u33...u3n......................000...unn]{displaystyle U=left[{begin{array}{ccccccccccccc}u_{11}{12}{12}{12}{13}{.}{1n}{1n}} fake.}{u_{22}{u.}{u_{23}{.{.}{1n.}{.{.}{.}{1n.}{.}{.}{.}{.}{.}{.
![{displaystyle U=left[{begin{array}{ccccccc}u_{11}&u_{12}&u_{13}&.&.&.&u_{1n}\0&u_{22}&u_{23}&.&.&.&u_{2n}\0&0&u_{33}&.&.&.&u_{3n}\.&.&..&.&.&.&.\.&.&.&.&.&.&.\.&.&.&.&.&.&.\0&0&0&.&.&.&u_{nn}\end{array}}right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1f3738aafefcfc6ce6672de166634ec192b17cb)
Similarly, a lower triangular matrix is said to be a matrix of the form:
- L=[chuckles]l1100...0l21l220...0l31l32l33...0.....................ln1ln2ln3...lnn]{displaystyle L=left[{begin{array}{ccccccccccc}l_{11}{11}{1}{1}{1}{1}{1}{1}{1}{1}{1}{1}{1}{1}{1}{1}{1}{1}{0}{1}{1}{3}{.}{.{1}{1st.}{1st.}{.}{1st.}{.}{1st.}{.}{1st.}{l.}{1st.}{1st.}{1st.}{1st.}{l.}{l.}{.}{.}{l.}{l.}{1st.}{.}{.}{1st.}{.}{.}{.}{.}{.}{.}{l.
![{displaystyle L=left[{begin{array}{ccccccc}l_{11}&0&0&.&.&.&0\l_{21}&l_{22}&0&.&.&.&0\l_{31}&l_{32}&l_{33}&.&.&.&0\.&.&.&.&.&.&.\.&.&.&.&.&.&.\.&.&.&.&.&.&.\l_{n1}&l_{n2}&l_{n3}&.&.&.&l_{nn}\end{array}}right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3037a19b38607b2023d249920dd48c19da110d02)
The letters U and L are usually used, respectively, since U is the initial of upper triangular matrix and L of lower triangular matrix, the names given to these matrices in english
Examples
This matrix is upper triangular:
- [chuckles]142034001]{displaystyle left[{begin{array}{ccc}1 supposed4 fake2\ fake3 nightmare4\ fake0 fake1end{array}}}{right]}}}
![{displaystyle left[{begin{array}{ccc}1&4&2\0&3&4\0&0&1end{array}}right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d192bca5095728c2d33cc91bdc93a434dd2d7da4)
This matrix is lower triangular:
- [chuckles]100280497]{displaystyle left[{begin{array}{ccc}1 fake0 hypo2 fake8 fake0\4 fake9 stranger7end{array}}}{right]}}
![{displaystyle left[{begin{array}{ccc}1&0&0\2&8&0\4&9&7end{array}}right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6202cbff611962073c80ee6c9ebde6044e0f365)
This matrix is unitary upper triangular, since the elements on its diagonal are 1, the unitary lower triangular matrix is analogous:
- [chuckles]142560148− − 1001430001300001]{displaystyle left[{begin{array}{ccc}{ccc}1 fake4}{4}{4}{}{}{ccc}1}{x4}{ccc}1}{4}{x1}1 fake4}{6}{ fake4}{ fake4}{ fake4}{1}{ fake4}{ fake4}{ fake4}{ fake4}{ fake4}{ fake4}{ fake4}{ fake4}{ fake4}{ fake4}{ fake4}{ fake4}{ fake4}{ fake4}{ fake3 fake4}{ fake4}{ fake4}{ fake4}{ fake4}{ fake3 fake4}{ fake4}{ fake4}{ fake4}{ fake4}{ fake4}{
![{displaystyle left[{begin{array}{ccc}1&4&2&5&6\0&1&4&8&-1\0&0&1&4&3\0&0&0&1&3\0&0&0&0&1\end{array}}right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51d2c7b6a079c85520fcba34d52f5f7f8ebca6cb)
Properties of triangular matrices
- A superior and lower diagonal triangular matrix in a base of own vectors (diagonal matrix) SI AND ONLY If the elements of your diagonal are different two to two.
- The product of two upper triangular matrices (inferiors) is a superior triangular matrix (inferior).
- The transposing of a higher triangular matrix is a lower triangular matrix and vice versa.
- The determinant of a triangular matrix is the product of the diagonal elements.
- A triangular matrix is invertible if and only if all the elements of the diagonal are not null. In this case, the reverse of a higher triangular matrix (inferior) is another upper triangular matrix (inferior).
- The values of a triangular matrix are the elements of the main diagonal.
Applications
By solving linear equation systems of the form Ax=b{displaystyle Ax=b}, it is common to use some factoring method to break down matrix A in factors such that simplify or facilitate the solution of the system, we have for example the LU factoring that breaks down A into two triangular matrices, one lower (Lower) L{displaystyle L}and another superior (Upper) U{displaystyle U}, such that A=LU{displaystyle A=LU}, obtaining the equivalent system LUx=b{displaystyle LUx=b}that replaced Ux{displaystyle Ux} for and{displaystyle and} can be solved separately Land=b{displaystyle Ly=b} And then stop. Ux=and{displaystyle Ux=y}.
A system of linear equations in matrix form
- Lx=b{displaystyle mathbf {L} mathbf {x} =mathbf {b}}}
or
- Ux=b{displaystyle mathbf {U} mathbf {x} =mathbf {b}}}
is very easy to solve. The first system can be written as
- l1,1x1=b1l2,1x1+l2,2x2=b2 lm,1x1+lm,2x2+...... +lm,mxm=bm{displaystyle {begin{matrix}l_{1,1}x_{1}{1}{1}{1l_{2,1}x_{1}{1}{1}{1}{2}{2}{xx_{2}{2}{2}{2}{2}{2}{2}{s}{xxxxxx,}{lxxxxxxxxxxxx?
which can be solved by following a simple recursive algorithm
- x1=b1l1,1,{displaystyle x_{1}={frac {b_{1}}{l_{1,1}}}}}}},
- x2=b2− − l2,1x1l2,2,{displaystyle x_{2}={frac {b_{2}-l_{2,1}x_{1}}}{l_{2,2}}}}},
- {displaystyle vdots }
- xm=bm− − ␡ ␡ i=1m− − 1lm,ixilm,m.{displaystyle x_{m}={frac {b_{m}-sum _{i=1}{m}-1l_{m,i}x_{i}}{l_{m,m}}}}}}}}} !
In an analogous way, a system given by an upper triangular matrix can be solved.
Cholesky factorization and LDLT factorization are other methods to decompose matrix A into triangular matrices, although these require matrix A to be symmetric and additionally for Cholesky factorization to be positive definite.