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Block matrix

From Wikipedia, the free encyclopedia
Matrix defined using smaller matrices called blocks

Inmathematics, ablock matrix or apartitioned matrix is amatrix that is interpreted as having been broken into sections calledblocks orsubmatrices.[1][2]

Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, orpartition it, into a collection of smaller matrices.[3][2] For example, the 3×4 matrix presented below is divided by horizontal and vertical lines into four blocks: the top-left 2×3 block, the top-right 2×1 block, the bottom-left 1×3 block, and the bottom-right 1×1 block.

[a11a12a13b1a21a22a23b2c1c2c3d]{\displaystyle \left[{\begin{array}{ccc|c}a_{11}&a_{12}&a_{13}&b_{1}\\a_{21}&a_{22}&a_{23}&b_{2}\\\hline c_{1}&c_{2}&c_{3}&d\end{array}}\right]}

Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned.

This notion can be made more precise for ann{\displaystyle n} bym{\displaystyle m} matrixM{\displaystyle M} by partitioningn{\displaystyle n} into a collectionrowgroups{\displaystyle {\text{rowgroups}}}, and then partitioningm{\displaystyle m} into a collectioncolgroups{\displaystyle {\text{colgroups}}}. The original matrix is then considered as the "total" of these groups, in the sense that the(i,j){\displaystyle (i,j)} entry of the original matrix corresponds in a1-to-1 way with some(s,t){\displaystyle (s,t)}offset entry of some(x,y){\displaystyle (x,y)}, wherex∈rowgroups{\displaystyle x\in {\text{rowgroups}}} andy∈colgroups{\displaystyle y\in {\text{colgroups}}}.[4]

Block matrix algebra arises in general frombiproducts incategories of matrices.[5]

A 168×168 element block matrix with 12×12, 12×24, 24×12, and 24×24 sub-matrices. Non-zero elements are in blue, zero elements are grayed.

Example

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The matrix

P=[1227156233453367]{\displaystyle \mathbf {P} ={\begin{bmatrix}1&2&2&7\\1&5&6&2\\3&3&4&5\\3&3&6&7\end{bmatrix}}}

can be visualized as divided into four blocks, as

P=[1227156233453367].{\displaystyle \mathbf {P} =\left[{\begin{array}{cc|cc}1&2&2&7\\1&5&6&2\\\hline 3&3&4&5\\3&3&6&7\end{array}}\right].}

The horizontal and vertical lines have no special mathematical meaning,[6][7] but are a common way to visualize a partition.[6][7] By this partition,P{\displaystyle P} is partitioned into four 2×2 blocks, as

P11=[1215],P12=[2762],P21=[3333],P22=[4567].{\displaystyle {\begin{aligned}\mathbf {P} _{11}&={\begin{bmatrix}1&2\\1&5\end{bmatrix}},&\mathbf {P} _{12}&={\begin{bmatrix}2&7\\6&2\end{bmatrix}},\\[1ex]\mathbf {P} _{21}&={\begin{bmatrix}3&3\\3&3\end{bmatrix}},&\mathbf {P} _{22}&={\begin{bmatrix}4&5\\6&7\end{bmatrix}}.\end{aligned}}}

The partitioned matrix can then be written as[8]

P=[P11P12P21P22].{\displaystyle \mathbf {P} ={\begin{bmatrix}\mathbf {P} _{11}&\mathbf {P} _{12}\\\mathbf {P} _{21}&\mathbf {P} _{22}\end{bmatrix}}.}

Formal definition

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LetA∈Cm×n{\displaystyle A\in \mathbb {C} ^{m\times n}}. Apartitioning ofA{\displaystyle A} is a representation ofA{\displaystyle A} in the form

A=[A11A12⋯A1qA21A22⋯A2q⋮⋮⋱⋮Ap1Ap2⋯Apq],{\displaystyle A={\begin{bmatrix}A_{11}&A_{12}&\cdots &A_{1q}\\A_{21}&A_{22}&\cdots &A_{2q}\\\vdots &\vdots &\ddots &\vdots \\A_{p1}&A_{p2}&\cdots &A_{pq}\end{bmatrix}},}

whereAij∈Cmi×nj{\displaystyle A_{ij}\in \mathbb {C} ^{m_{i}\times n_{j}}} are contiguous submatrices,∑i=1pmi=m{\textstyle \sum _{i=1}^{p}m_{i}=m}, and∑j=1qnj=n{\textstyle \sum _{j=1}^{q}n_{j}=n}.[9] The elementsAij{\displaystyle A_{ij}} of the partition are calledblocks.[9]

By this definition, the blocks in any one column must all have the same number of columns.[9] Similarly, the blocks in any one row must have the same number of rows.[9]

Partitioning methods

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A matrix can be partitioned in many ways.[9] For example, a matrixA{\displaystyle A} is said to bepartitioned by columns if it is written as

A=(a1 a2 ⋯ an),{\displaystyle A=(a_{1}\ a_{2}\ \cdots \ a_{n}),}

whereaj{\displaystyle a_{j}} is thej{\displaystyle j}th column ofA{\displaystyle A}.[9] A matrix can also bepartitioned by rows:

A=[a1Ta2T⋮amT],{\displaystyle A={\begin{bmatrix}a_{1}^{T}\\a_{2}^{T}\\\vdots \\a_{m}^{T}\end{bmatrix}},}

whereaiT{\displaystyle a_{i}^{T}} is thei{\displaystyle i}-th row ofA{\displaystyle A}.[9]

Common partitions

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Often,[9] we encounter the 2×2 partition

A=[A11A12A21A22],{\displaystyle A={\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{bmatrix}},}

particularly in the form whereA11{\displaystyle A_{11}} is a scalar:[9]

A=[a11a12Ta21A22].{\displaystyle A={\begin{bmatrix}a_{11}&a_{12}^{T}\\a_{21}&A_{22}\end{bmatrix}}.}

Block matrix operations

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Transpose

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Let

A=[A11A12⋯A1qA21A22⋯A2q⋮⋮⋱⋮Ap1Ap2⋯Apq]{\displaystyle A={\begin{bmatrix}A_{11}&A_{12}&\cdots &A_{1q}\\A_{21}&A_{22}&\cdots &A_{2q}\\\vdots &\vdots &\ddots &\vdots \\A_{p1}&A_{p2}&\cdots &A_{pq}\end{bmatrix}}}

whereAij∈Cki×ℓj{\displaystyle A_{ij}\in \mathbb {C} ^{k_{i}\times \ell _{j}}}. (This matrixA{\displaystyle A} will be reused in§ Addition and§ Multiplication.) Then its transpose is[9][10]

AT=[A11TA21T⋯Ap1TA12TA22T⋯Ap2T⋮⋮⋱⋮A1qTA2qT⋯ApqT],{\displaystyle A^{T}={\begin{bmatrix}A_{11}^{T}&A_{21}^{T}&\cdots &A_{p1}^{T}\\A_{12}^{T}&A_{22}^{T}&\cdots &A_{p2}^{T}\\\vdots &\vdots &\ddots &\vdots \\A_{1q}^{T}&A_{2q}^{T}&\cdots &A_{pq}^{T}\end{bmatrix}},}

and the same equation holds with the transpose replaced by the conjugate transpose.[9]

Block transpose

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A special form of matrixtranspose can also be defined for block matrices, where individual blocks are reordered but not transposed. LetA=(Bij){\displaystyle A=(B_{ij})} be ak×l{\displaystyle k\times l} block matrix withm×n{\displaystyle m\times n} blocksBij{\displaystyle B_{ij}}, the block transpose ofA{\displaystyle A} is thel×k{\displaystyle l\times k} block matrixAB{\displaystyle A^{\mathcal {B}}} withm×n{\displaystyle m\times n} blocks(AB)ij=Bji{\displaystyle \left(A^{\mathcal {B}}\right)_{ij}=B_{ji}}.[11] As with the conventional trace operator, the block transpose is alinear mapping such that(A+C)B=AB+CB{\displaystyle (A+C)^{\mathcal {B}}=A^{\mathcal {B}}+C^{\mathcal {B}}}.[10] However, in general the property(AC)B=CBAB{\displaystyle (AC)^{\mathcal {B}}=C^{\mathcal {B}}A^{\mathcal {B}}} does not hold unless the blocks ofA{\displaystyle A} andC{\displaystyle C} commute.

Addition

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Let

B=[B11B12⋯B1sB21B22⋯B2s⋮⋮⋱⋮Br1Br2⋯Brs],{\displaystyle B={\begin{bmatrix}B_{11}&B_{12}&\cdots &B_{1s}\\B_{21}&B_{22}&\cdots &B_{2s}\\\vdots &\vdots &\ddots &\vdots \\B_{r1}&B_{r2}&\cdots &B_{rs}\end{bmatrix}},}

whereBij∈Cmi×nj{\displaystyle B_{ij}\in \mathbb {C} ^{m_{i}\times n_{j}}}, and letA{\displaystyle A} be the matrix defined in§ Transpose. (This matrixB{\displaystyle B} will be reused in§ Multiplication.) Then ifp=r{\displaystyle p=r},q=s{\displaystyle q=s},ki=mi{\displaystyle k_{i}=m_{i}}, andℓj=nj{\displaystyle \ell _{j}=n_{j}}, then[9]

A+B=[A11+B11A12+B12⋯A1q+B1qA21+B21A22+B22⋯A2q+B2q⋮⋮⋱⋮Ap1+Bp1Ap2+Bp2⋯Apq+Bpq].{\displaystyle A+B={\begin{bmatrix}A_{11}+B_{11}&A_{12}+B_{12}&\cdots &A_{1q}+B_{1q}\\A_{21}+B_{21}&A_{22}+B_{22}&\cdots &A_{2q}+B_{2q}\\\vdots &\vdots &\ddots &\vdots \\A_{p1}+B_{p1}&A_{p2}+B_{p2}&\cdots &A_{pq}+B_{pq}\end{bmatrix}}.}

Multiplication

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It is possible to use a block partitioned matrix product that involves only algebra on submatrices of the factors. The partitioning of the factors is not arbitrary, however, and requires "conformable partitions"[12] between two matricesA{\displaystyle A} andB{\displaystyle B} such that all submatrix products that will be used are defined.[13]

Two matricesA{\displaystyle A} andB{\displaystyle B} are said to be partitioned conformally for the productAB{\displaystyle AB}, whenA{\displaystyle A} andB{\displaystyle B} are partitioned into submatrices and if the multiplicationAB{\displaystyle AB} is carried out treating the submatrices as if they are scalars, but keeping the order, and when all products and sums of submatrices involved are defined.

— Arak M. Mathai and Hans J. Haubold,Linear Algebra: A Course for Physicists and Engineers[14]

LetA{\displaystyle A} be the matrix defined in§ Transpose, and letB{\displaystyle B} be the matrix defined in§ Addition. Then the matrix product

C=AB{\displaystyle C=AB}

can be performed blockwise, yieldingC{\displaystyle C} as an(p×s){\displaystyle (p\times s)} matrix. The matrices in the resulting matrixC{\displaystyle C} are calculated by multiplying:[6]

Cij=∑k=1qAikBkj.{\displaystyle C_{ij}=\sum _{k=1}^{q}A_{ik}B_{kj}.}

Or, using theEinstein notation that implicitly sums over repeated indices:

Cij=AikBkj.{\displaystyle C_{ij}=A_{ik}B_{kj}.}

DepictingC{\displaystyle C} as a matrix, we have[9]

C=AB=[∑i=1qA1iBi1∑i=1qA1iBi2⋯∑i=1qA1iBis∑i=1qA2iBi1∑i=1qA2iBi2⋯∑i=1qA2iBis⋮⋮⋱⋮∑i=1qApiBi1∑i=1qApiBi2⋯∑i=1qApiBis].{\displaystyle C=AB={\begin{bmatrix}\sum _{i=1}^{q}A_{1i}B_{i1}&\sum _{i=1}^{q}A_{1i}B_{i2}&\cdots &\sum _{i=1}^{q}A_{1i}B_{is}\\\sum _{i=1}^{q}A_{2i}B_{i1}&\sum _{i=1}^{q}A_{2i}B_{i2}&\cdots &\sum _{i=1}^{q}A_{2i}B_{is}\\\vdots &\vdots &\ddots &\vdots \\\sum _{i=1}^{q}A_{pi}B_{i1}&\sum _{i=1}^{q}A_{pi}B_{i2}&\cdots &\sum _{i=1}^{q}A_{pi}B_{is}\end{bmatrix}}.}

Inversion

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For more details and derivation using block LDU decomposition, seeSchur complement.
See also:Helmert–Wolf blocking

If a matrix is partitioned into four blocks, it can beinverted blockwise as follows:

P=[ABCD]−1=[A−1+A−1B(D−CA−1B)−1CA−1−A−1B(D−CA−1B)−1−(D−CA−1B)−1CA−1(D−CA−1B)−1],{\displaystyle {\begin{aligned}P&={\begin{bmatrix}A&B\\C&D\end{bmatrix}}^{-1}\\[1ex]&={\begin{bmatrix}A^{-1}+A^{-1}B\left(D-CA^{-1}B\right)^{-1}CA^{-1}&-A^{-1}B\left(D-CA^{-1}B\right)^{-1}\\-\left(D-CA^{-1}B\right)^{-1}CA^{-1}&\left(D-CA^{-1}B\right)^{-1}\end{bmatrix}},\end{aligned}}}

whereA andD are square blocks of arbitrary size, andB andC areconformable with them for partitioning. Furthermore,A and the Schur complement ofA inP:P/A =D −CA−1B must be invertible.[15]

Equivalently, by permuting the blocks:[16]

P=[ABCD]−1=[(A−BD−1C)−1−(A−BD−1C)−1BD−1−D−1C(A−BD−1C)−1D−1+D−1C(A−BD−1C)−1BD−1].{\displaystyle {\begin{aligned}P&={\begin{bmatrix}A&B\\C&D\end{bmatrix}}^{-1}\\[1ex]&={\begin{bmatrix}\left(A-BD^{-1}C\right)^{-1}&-\left(A-BD^{-1}C\right)^{-1}BD^{-1}\\-D^{-1}C\left(A-BD^{-1}C\right)^{-1}&D^{-1}+D^{-1}C\left(A-BD^{-1}C\right)^{-1}BD^{-1}\end{bmatrix}}.\end{aligned}}}

Here,D and the Schur complement ofD inP:P/D =A −BD−1C must be invertible.

IfA andD are both invertible, then:

[ABCD]−1=[(A−BD−1C)−100(D−CA−1B)−1][I−BD−1−CA−1I].{\displaystyle {\begin{bmatrix}A&B\\C&D\end{bmatrix}}^{-1}={\begin{bmatrix}\left(A-BD^{-1}C\right)^{-1}&0\\0&\left(D-CA^{-1}B\right)^{-1}\end{bmatrix}}{\begin{bmatrix}I&-BD^{-1}\\-CA^{-1}&I\end{bmatrix}}.}

By theWeinstein–Aronszajn identity, one of the two matrices in the block-diagonal matrix is invertible exactly when the other is.

Computing submatrix inverses from the full inverse

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By the symmetry between a matrix and its inverse in the block inversion formula, if a matrixP and its inverseP−1 are partitioned conformally:

P=[ABCD],P−1=[EFGH]{\displaystyle P={\begin{bmatrix}{A}&{B}\\{C}&{D}\end{bmatrix}},\quad P^{-1}={\begin{bmatrix}{E}&{F}\\{G}&{H}\end{bmatrix}}}

then the inverse of any principal submatrix can be computed from the corresponding blocks ofP−1:

A−1=E−FH−1G{\displaystyle {A}^{-1}={E}-{FH}^{-1}{G}}D−1=H−GE−1F{\displaystyle {D}^{-1}={H}-{GE}^{-1}{F}}

This relationship follows from recognizing thatE−1 =A −BD−1C (the Schur complement), and applying the same block inversion formula with the roles ofP andP−1 reversed.[17][18]

Determinant

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The formula for the determinant of a2×2{\displaystyle 2\times 2}-matrix above continues to hold, under appropriate further assumptions, for a matrix composed of four submatricesA,B,C,D{\displaystyle A,B,C,D} withA{\displaystyle A} andD{\displaystyle D} square. The easiest such formula, which can be proven using either theLeibniz formula or a factorization involving theSchur complement, is[16]det[A0CD]=det(A)det(D)=det[AB0D].{\displaystyle \det {\begin{bmatrix}A&0\\C&D\end{bmatrix}}=\det(A)\det(D)=\det {\begin{bmatrix}A&B\\0&D\end{bmatrix}}.}

Using this formula, we can derive thatcharacteristic polynomials of[A0CD]{\displaystyle {\begin{bmatrix}A&0\\C&D\end{bmatrix}}} and[AB0D]{\displaystyle {\begin{bmatrix}A&B\\0&D\end{bmatrix}}} are same and equal to the product of characteristic polynomials ofA{\displaystyle A} andD{\displaystyle D}. Furthermore, If[A0CD]{\displaystyle {\begin{bmatrix}A&0\\C&D\end{bmatrix}}} or[AB0D]{\displaystyle {\begin{bmatrix}A&B\\0&D\end{bmatrix}}} isdiagonalizable, thenA{\displaystyle A} andD{\displaystyle D} are diagonalizable too. The converse is false; simply check[1101]{\displaystyle {\begin{bmatrix}1&1\\0&1\end{bmatrix}}}.

IfA{\displaystyle A} isinvertible, one has[16]

det[ABCD]=det(A)det(D−CA−1B),{\displaystyle \det {\begin{bmatrix}A&B\\C&D\end{bmatrix}}=\det(A)\det \left(D-CA^{-1}B\right),}

and ifD{\displaystyle D} is invertible, one has[19][16]

det[ABCD]=det(D)det(A−BD−1C).{\displaystyle \det {\begin{bmatrix}A&B\\C&D\end{bmatrix}}=\det(D)\det \left(A-BD^{-1}C\right).}

If the blocks are square matrices of thesame size further formulas hold. For example, ifC{\displaystyle C} andD{\displaystyle D}commute (i.e.,CD=DC{\displaystyle CD=DC}), then[20]det[ABCD]=det(AD−BC).{\displaystyle \det {\begin{bmatrix}A&B\\C&D\end{bmatrix}}=\det(AD-BC).}Similar statements hold whenAB=BA{\displaystyle AB=BA},AC=CA{\displaystyle AC=CA}, or⁠BD=DB{\displaystyle BD=DB}⁠. Namely, ifAC=CA{\displaystyle AC=CA}, thendet[ABCD]=det(AD−CB).{\displaystyle \det {\begin{bmatrix}A&B\\C&D\end{bmatrix}}=\det(AD-CB).}Note the change in order ofC{\displaystyle C} andB{\displaystyle B} (we haveCB{\displaystyle CB} instead ofBC{\displaystyle BC}). Similarly, ifBD=DB{\displaystyle BD=DB}, thenAD{\displaystyle AD} should be replaced withDA{\displaystyle DA} (i.e. we getdet(DA−BC){\displaystyle \det(DA-BC)}) and ifAB=BA{\displaystyle AB=BA}, then we should havedet(DA−CB){\displaystyle \det(DA-CB)}. Note for the last two results, you have to use commutativity of the underlying ring, but not for the first two.

This formula has been generalized to matrices composed of more than2×2{\displaystyle 2\times 2} blocks, again under appropriate commutativity conditions among the individual blocks.[21]

ForA=D{\displaystyle A=D} andB=C{\displaystyle B=C}, the following formula holds (even ifA{\displaystyle A} andB{\displaystyle B} do not commute)[16]det[ABBA]=det(A−B)det(A+B).{\displaystyle \det {\begin{bmatrix}A&B\\B&A\end{bmatrix}}=\det(A-B)\det(A+B).}

Special types of block matrices

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Direct sums and block diagonal matrices

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Direct sum

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See also:Direct sum of matrices

For any arbitrary matricesA (of sizem × n) andB (of sizep × q), we have thedirect sum ofA andB, denoted byA ⊕ B and defined as[10]

A⊕B=[a11⋯a1n0⋯0⋮⋱⋮⋮⋱⋮am1⋯amn0⋯00⋯0b11⋯b1q⋮⋱⋮⋮⋱⋮0⋯0bp1⋯bpq].{\displaystyle {A}\oplus {B}={\begin{bmatrix}a_{11}&\cdots &a_{1n}&0&\cdots &0\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\a_{m1}&\cdots &a_{mn}&0&\cdots &0\\0&\cdots &0&b_{11}&\cdots &b_{1q}\\\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\0&\cdots &0&b_{p1}&\cdots &b_{pq}\end{bmatrix}}.}

For instance,

[132231]⊕[1601]=[13200231000001600001].{\displaystyle {\begin{bmatrix}1&3&2\\2&3&1\end{bmatrix}}\oplus {\begin{bmatrix}1&6\\0&1\end{bmatrix}}={\begin{bmatrix}1&3&2&0&0\\2&3&1&0&0\\0&0&0&1&6\\0&0&0&0&1\end{bmatrix}}.}

This operation generalizes naturally to arbitrary dimensioned arrays (provided thatA andB have the same number of dimensions).

Note that any element in thedirect sum of twovector spaces of matrices could be represented as a direct sum of two matrices.

Block diagonal matrices

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See also:Diagonal matrix

Ablock diagonal matrix is a block matrix that is asquare matrix such that the main-diagonal blocks are square matrices and all off-diagonal blocks are zero matrices.[16] That is, a block diagonal matrixA has the form

A=[A10⋯00A2⋯0⋮⋮⋱⋮00⋯An]{\displaystyle {A}={\begin{bmatrix}A_{1}&0&\cdots &0\\0&A_{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &A_{n}\end{bmatrix}}}

whereAk is a square matrix for allk = 1, ...,n. In other words, matrixA is thedirect sum ofA1, ...,An.[16] It can also be indicated asA1 ⊕ A2 ⊕ ... ⊕ An[10] or diag(A1,A2, ...,An)[10] (the latter being the same formalism used for adiagonal matrix). Any square matrix can trivially be considered a block diagonal matrix with only one block.

For thedeterminant andtrace, the following properties hold:

detA=detA1×⋯×detAn,{\displaystyle {\begin{aligned}\det {A}&=\det {A}_{1}\times \cdots \times \det {A}_{n},\end{aligned}}}[22][23] and
tr⁡A=tr⁡A1+⋯+tr⁡An.{\displaystyle {\begin{aligned}\operatorname {tr} {A}&=\operatorname {tr} {A}_{1}+\cdots +\operatorname {tr} {A}_{n}.\end{aligned}}}[16][23]

A block diagonal matrix is invertibleif and only if each of its main-diagonal blocks are invertible, and in this case its inverse is another block diagonal matrix given by[24][A10⋯00A2⋯0⋮⋮⋱⋮00⋯An]−1=[A1−10⋯00A2−1⋯0⋮⋮⋱⋮00⋯An−1].{\displaystyle {\begin{bmatrix}{A}_{1}&{0}&\cdots &{0}\\{0}&{A}_{2}&\cdots &{0}\\\vdots &\vdots &\ddots &\vdots \\{0}&{0}&\cdots &{A}_{n}\end{bmatrix}}^{-1}={\begin{bmatrix}{A}_{1}^{-1}&{0}&\cdots &{0}\\{0}&{A}_{2}^{-1}&\cdots &{0}\\\vdots &\vdots &\ddots &\vdots \\{0}&{0}&\cdots &{A}_{n}^{-1}\end{bmatrix}}.}

Theeigenvalues[25]and eigenvectors ofA{\displaystyle {A}} are simply those of theAk{\displaystyle {A}_{k}}s combined.[23]

Block tridiagonal matrices

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See also:Tridiagonal matrix

Ablock tridiagonal matrix is another special block matrix, which is just like the block diagonal matrix asquare matrix, having square matrices (blocks) in the lower diagonal,main diagonal and upper diagonal, with all other blocks being zero matrices. It is essentially atridiagonal matrix but has submatrices in places of scalars. A block tridiagonal matrixA{\displaystyle A} has the form

A=[B1C1⋯0A2B2C2⋱⋱⋱⋮AkBkCk⋮⋱⋱⋱An−1Bn−1Cn−10⋯AnBn]{\displaystyle {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}}}

whereAk{\displaystyle {A}_{k}},Bk{\displaystyle {B}_{k}} andCk{\displaystyle {C}_{k}} are square sub-matrices of the lower, main and upper diagonal respectively.[26][27]

Block tridiagonal matrices are often encountered in numerical solutions of engineering problems (e.g.,computational fluid dynamics). Optimized numerical methods forLU factorization are available[28] and hence efficient solution algorithms for equation systems with a block tridiagonal matrix as coefficient matrix. TheThomas algorithm, used for efficient solution of equation systems involving atridiagonal matrix can also be applied using matrix operations to block tridiagonal matrices (see alsoBlock LU decomposition).

Block triangular matrices

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See also:Triangular matrix

Ann×n{\displaystyle n\times n} matrixA{\displaystyle A} isupper block triangular (orblock upper triangular[29]) if there are positive integersn1,…,nk{\displaystyle n_{1},\ldots ,n_{k}} such thatn=n1+n2+…+nk{\displaystyle n=n_{1}+n_{2}+\ldots +n_{k}} andA=[A11A12⋯A1k0A22⋯A2k⋮⋮⋱⋮00⋯Akk],{\displaystyle A={\begin{bmatrix}A_{11}&A_{12}&\cdots &A_{1k}\\0&A_{22}&\cdots &A_{2k}\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &A_{kk}\end{bmatrix}},}where the matrixAij{\displaystyle A_{ij}} isni×nj{\displaystyle n_{i}\times n_{j}} for alli,j=1,…,k{\displaystyle i,j=1,\ldots ,k}.[25][29]Similarly,A{\displaystyle A} islower block triangular ifA=[A110⋯0A21A22⋯0⋮⋮⋱⋮Ak1Ak2⋯Akk],{\displaystyle A={\begin{bmatrix}A_{11}&0&\cdots &0\\A_{21}&A_{22}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\A_{k1}&A_{k2}&\cdots &A_{kk}\end{bmatrix}},}whereAij{\displaystyle A_{ij}} isni×nj{\displaystyle n_{i}\times n_{j}} for alli,j=1,…,k{\displaystyle i,j=1,\ldots ,k}.[25]

Block Toeplitz matrices

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See also:Toeplitz matrix

Ablock Toeplitz matrix is another special block matrix, which contains blocks that are repeated down the diagonals of the matrix, as aToeplitz matrix has elements repeated down the diagonal.

A matrixA{\displaystyle A} isblock Toeplitz ifA(i,j)=A(k,l){\displaystyle A_{(i,j)}=A_{(k,l)}} for allk−i=l−j{\displaystyle k-i=l-j}, that is,

A=[A1A2A3⋯A4A1A2⋯A5A4A1⋯⋮⋮⋮⋱],{\displaystyle A={\begin{bmatrix}A_{1}&A_{2}&A_{3}&\cdots \\A_{4}&A_{1}&A_{2}&\cdots \\A_{5}&A_{4}&A_{1}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{bmatrix}},}

whereAi∈Fni×mi{\displaystyle A_{i}\in \mathbb {F} ^{n_{i}\times m_{i}}}.[25]

Block Hankel matrices

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See also:Hankel matrix

A matrixA{\displaystyle A} isblock Hankel ifA(i,j)=A(k,l){\displaystyle A_{(i,j)}=A_{(k,l)}} for alli+j=k+l{\displaystyle i+j=k+l}, that is,

A=[A1A2A3⋯A2A3A4⋯A3A4A5⋯⋮⋮⋮⋱],{\displaystyle A={\begin{bmatrix}A_{1}&A_{2}&A_{3}&\cdots \\A_{2}&A_{3}&A_{4}&\cdots \\A_{3}&A_{4}&A_{5}&\cdots \\\vdots &\vdots &\vdots &\ddots \end{bmatrix}},}

whereAi∈Fni×mi{\displaystyle A_{i}\in \mathbb {F} ^{n_{i}\times m_{i}}}.[25]

See also

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  • Kronecker product (matrix direct product resulting in a block matrix)
  • Jordan normal form (canonical form of a linear operator on a finite-dimensional complex vector space)
  • Strassen algorithm (algorithm for matrix multiplication that is faster than the conventional matrix multiplication algorithm)

Notes

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  1. ^Eves, Howard (1980).Elementary Matrix Theory (reprint ed.). New York: Dover. p. 37.ISBN 0-486-63946-0. Retrieved24 April 2013.We shall find that it is sometimes convenient to subdivide a matrix into rectangular blocks of elements. This leads us to consider so-calledpartitioned, orblock,matrices.
  2. ^abDobrushkin, Vladimir."Partition Matrices".Linear Algebra with Mathematica. Retrieved2024-03-24.
  3. ^Anton, Howard (1994).Elementary Linear Algebra (7th ed.). New York: John Wiley. p. 30.ISBN 0-471-58742-7.A matrix can be subdivided orpartitioned into smaller matrices by inserting horizontal and vertical rules between selected rows and columns.
  4. ^Indhumathi, D.; Sarala, S. (2014-05-16)."Fragment Analysis and Test Case Generation using F-Measure for Adaptive Random Testing and Partitioned Block based Adaptive Random Testing"(PDF).International Journal of Computer Applications.93 (6): 13.Bibcode:2014IJCA...93f..11I.doi:10.5120/16218-5662.
  5. ^Macedo, H.D.; Oliveira, J.N. (2013). "Typing linear algebra: A biproduct-oriented approach".Science of Computer Programming.78 (11):2160–2191.arXiv:1312.4818.doi:10.1016/j.scico.2012.07.012.
  6. ^abcJohnston, Nathaniel (2021).Introduction to linear and matrix algebra. Cham, Switzerland: Springer Nature. pp. 30, 425.ISBN 978-3-030-52811-9.
  7. ^abJohnston, Nathaniel (2021).Advanced linear and matrix algebra. Cham, Switzerland: Springer Nature. p. 298.ISBN 978-3-030-52814-0.
  8. ^Jeffrey, Alan (2010).Matrix operations for engineers and scientists: an essential guide in linear algebra. Dordrecht [Netherlands] ; New York: Springer. p. 54.ISBN 978-90-481-9273-1.OCLC 639165077.
  9. ^abcdefghijklmStewart, Gilbert W. (1998).Matrix algorithms. 1: Basic decompositions. Philadelphia, PA: Soc. for Industrial and Applied Mathematics. pp. 18–20.ISBN 978-0-89871-414-2.
  10. ^abcdeGentle, James E. (2007).Matrix Algebra: Theory, Computations, and Applications in Statistics. Springer Texts in Statistics. New York, NY: Springer New York Springer e-books. pp. 47, 487.ISBN 978-0-387-70873-7.
  11. ^Mackey, D. Steven (2006).Structured linearizations for matrix polynomials(PDF) (Thesis). University of Manchester.ISSN 1749-9097.OCLC 930686781.
  12. ^Eves, Howard (1980).Elementary Matrix Theory (reprint ed.). New York: Dover. p. 37.ISBN 0-486-63946-0. Retrieved24 April 2013.A partitioning as in Theorem 1.9.4 is called aconformable partition ofA andB.
  13. ^Anton, Howard (1994).Elementary Linear Algebra (7th ed.). New York: John Wiley. p. 36.ISBN 0-471-58742-7....provided the sizes of the submatrices of A and B are such that the indicated operations can be performed.
  14. ^Mathai, Arakaparampil M.; Haubold, Hans J. (2017).Linear Algebra: a course for physicists and engineers. De Gruyter textbook. Berlin Boston: De Gruyter. p. 162.ISBN 978-3-11-056259-0.
  15. ^Bernstein, Dennis (2005).Matrix Mathematics. Princeton University Press. p. 44.ISBN 0-691-11802-7.
  16. ^abcdefghAbadir, Karim M.; Magnus, Jan R. (2005).Matrix Algebra. Cambridge University Press. pp. 97, 100, 106, 111, 114, 118.ISBN 9781139443647.
  17. ^"Is this formula for a matrix block inverse in terms of the entire matrix inverse known?".MathOverflow.
  18. ^Escalante-B., Alberto N.; Wiskott, Laurenz (2016)."Improved graph-based SFA: Information preservation complements the slowness principle".Machine Learning.arXiv:1412.4679.doi:10.1007/s10994-016-5563-y.
  19. ^Taboga, Marco (2021). "Determinant of a block matrix", Lectures on matrix algebra.
  20. ^Silvester, J. R. (2000)."Determinants of Block Matrices"(PDF).Math. Gaz.84 (501):460–467.doi:10.2307/3620776.JSTOR 3620776. Archived fromthe original(PDF) on 2015-03-18. Retrieved2021-06-25.
  21. ^Sothanaphan, Nat (January 2017). "Determinants of block matrices with noncommuting blocks".Linear Algebra and Its Applications.512:202–218.arXiv:1805.06027.doi:10.1016/j.laa.2016.10.004.S2CID 119272194.
  22. ^Quarteroni, Alfio; Sacco, Riccardo; Saleri, Fausto (2000).Numerical mathematics. Texts in applied mathematics. New York: Springer. pp. 10, 13.ISBN 978-0-387-98959-4.
  23. ^abcGeorge, Raju K.; Ajayakumar, Abhijith (2024)."A Course in Linear Algebra".University Texts in the Mathematical Sciences: 35, 407.doi:10.1007/978-981-99-8680-4.ISBN 978-981-99-8679-8.ISSN 2731-9318.
  24. ^Prince, Simon J. D. (2012).Computer vision: models, learning, and inference. New York: Cambridge university press. p. 531.ISBN 978-1-107-01179-3.
  25. ^abcdeBernstein, Dennis S. (2009).Matrix mathematics: theory, facts, and formulas (2 ed.). Princeton, NJ: Princeton University Press. pp. 168, 298.ISBN 978-0-691-14039-1.
  26. ^Dietl, Guido K. E. (2007).Linear estimation and detection in Krylov subspaces. Foundations in signal processing, communications and networking. Berlin ; New York: Springer. pp. 85, 87.ISBN 978-3-540-68478-7.OCLC 85898525.
  27. ^Horn, Roger A.; Johnson, Charles R. (2017).Matrix analysis (Second edition, corrected reprint ed.). New York, NY: Cambridge University Press. p. 36.ISBN 978-0-521-83940-2.
  28. ^Datta, Biswa Nath (2010).Numerical linear algebra and applications (2 ed.). Philadelphia, Pa: SIAM. p. 168.ISBN 978-0-89871-685-6.
  29. ^abStewart, Gilbert W. (2001).Matrix algorithms. 2: Eigensystems. Philadelphia, Pa: Soc. for Industrial and Applied Mathematics. p. 5.ISBN 978-0-89871-503-3.

References

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Linear equations
Three dimensional Euclidean space
Matrices
Matrix decompositions
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Vector spaces
Structures
Multilinear algebra
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