Editing Block matrix (section)

===Block tridiagonal matrices===
{{See also|Tridiagonal matrix}}
A '''block tridiagonal matrix''' is another special block matrix, which is just like the block diagonal matrix a [[square matrix]], having square matrices (blocks) in the lower diagonal, [[main diagonal]] and upper diagonal, with all other blocks being zero matrices. It is essentially a [[tridiagonal matrix]] but has submatrices in places of scalars. A block tridiagonal matrix <math>A</math> has the form

:<math>{A} = \begin{bmatrix}
  {B}_{1} & {C}_{1} &                &                &           \cdots &                  &      {0}  \\
  {A}_{2} & {B}_{2} & {C}_{2} &                &                  &                  &                  \\
                 &         \ddots &         \ddots &         \ddots &                  &                  &           \vdots \\
                 &                & {A}_{k} & {B}_{k} &   {C}_{k} &                  &                  \\
          \vdots &                &                &         \ddots &           \ddots &           \ddots &                  \\
                 &                &                &                & {A}_{n-1} & {B}_{n-1} & {C}_{n-1} \\
      {0} &                &         \cdots &                &                  &   {A}_{n} &   {B}_{n}
\end{bmatrix}</math>

where <math>{A}_{k}</math>, <math>{B}_{k}</math> and <math>{C}_{k}</math> are square sub-matrices of the lower, main and upper diagonal respectively.<ref>{{Cite book |last=Dietl |first=Guido K. E. |url=https://www.worldcat.org/title/ocm85898525 |title=Linear estimation and detection in Krylov subspaces |date=2007 |publisher=Springer |isbn=978-3-540-68478-7 |series=Foundations in signal processing, communications and networking |location=Berlin ; New York |pages=85,87 |language=en |oclc=ocm85898525}}</ref><ref>{{Cite book |last1=Horn |first1=Roger A. |title=Matrix analysis |last2=Johnson |first2=Charles R. |date=2017 |publisher=Cambridge University Press |isbn=978-0-521-83940-2 |edition=Second edition, corrected reprint |location=New York, NY |pages=36 |language=en}}</ref>

Block tridiagonal matrices are often encountered in numerical solutions of engineering problems (e.g., [[computational fluid dynamics]]). Optimized numerical methods for [[LU factorization]] are available<ref>{{Cite book |last=Datta |first=Biswa Nath |title=Numerical linear algebra and applications |date=2010 |publisher=SIAM |isbn=978-0-89871-685-6 |edition=2 |location=Philadelphia, Pa |pages=168}}</ref> and hence efficient solution algorithms for equation systems with a block tridiagonal matrix as coefficient matrix. The [[Thomas algorithm]], used for efficient solution of equation systems involving a [[tridiagonal matrix]] can also be applied using matrix operations to block tridiagonal matrices (see also [[Block LU decomposition]]).