# Block matrix

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Matrix defined using smaller matrices called blocks

In [mathematics](/source/Mathematics), a **block matrix** or a **partitioned matrix** is a [matrix](/source/Matrix_(mathematics)) that is interpreted as having been broken into sections called **blocks** or **submatrices**.[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, or [partition](/source/Partition_of_a_set) 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.

[ a 11 a 12 a 13 b 1 a 21 a 22 a 23 b 2 c 1 c 2 c 3 d ] {\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 an n {\displaystyle n} by m {\displaystyle m} matrix M {\displaystyle M} by partitioning n {\displaystyle n} into a collection rowgroups {\displaystyle {\text{rowgroups}}} , and then partitioning m {\displaystyle m} into a collection colgroups {\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 a [1-to-1](/source/Bijection) way with some ( s , t ) {\displaystyle (s,t)} [offset](/source/Offset_(computer_science)) entry of some ( x , y ) {\displaystyle (x,y)} , where x ∈ rowgroups {\displaystyle x\in {\text{rowgroups}}} and y ∈ colgroups {\displaystyle y\in {\text{colgroups}}} .[4]

Block matrix algebra arises in general from [biproducts](/source/Biproduct) in [categories](/source/Category_(mathematics)) 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

The matrix

P = [ 1 2 2 7 1 5 6 2 3 3 4 5 3 3 6 7 ] {\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 = [ 1 2 2 7 1 5 6 2 3 3 4 5 3 3 6 7 ] . {\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

P 11 = [ 1 2 1 5 ] , P 12 = [ 2 7 6 2 ] , P 21 = [ 3 3 3 3 ] , P 22 = [ 4 5 6 7 ] . {\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 = [ P 11 P 12 P 21 P 22 ] . {\displaystyle \mathbf {P} ={\begin{bmatrix}\mathbf {P} _{11}&\mathbf {P} _{12}\\\mathbf {P} _{21}&\mathbf {P} _{22}\end{bmatrix}}.}

## Formal definition

Let A ∈ C m × n {\displaystyle A\in \mathbb {C} ^{m\times n}} . A **partitioning** of A {\displaystyle A} is a representation of A {\displaystyle A} in the form

A = [ A 11 A 12 ⋯ A 1 q A 21 A 22 ⋯ A 2 q ⋮ ⋮ ⋱ ⋮ A p 1 A p 2 ⋯ A p q ] , {\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}},}

where A i j ∈ C m i × n j {\displaystyle A_{ij}\in \mathbb {C} ^{m_{i}\times n_{j}}} are contiguous submatrices, ∑ i = 1 p m i = m {\textstyle \sum _{i=1}^{p}m_{i}=m} , and ∑ j = 1 q n j = n {\textstyle \sum _{j=1}^{q}n_{j}=n} .[9] The elements A i j {\displaystyle A_{ij}} of the partition are called **blocks**.[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

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

A = ( a 1 a 2 ⋯ a n ) , {\displaystyle A=(a_{1}\ a_{2}\ \cdots \ a_{n}),}

where a j {\displaystyle a_{j}} is the j {\displaystyle j} th column of A {\displaystyle A} .[9] A matrix can also be **partitioned by rows**:

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

where a i T {\displaystyle a_{i}^{T}} is the i {\displaystyle i} -th row of A {\displaystyle A} .[9]

### Common partitions

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

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

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

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

## Block matrix operations

### Transpose

Let

A = [ A 11 A 12 ⋯ A 1 q A 21 A 22 ⋯ A 2 q ⋮ ⋮ ⋱ ⋮ A p 1 A p 2 ⋯ A p q ] {\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}}}

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

A T = [ A 11 T A 21 T ⋯ A p 1 T A 12 T A 22 T ⋯ A p 2 T ⋮ ⋮ ⋱ ⋮ A 1 q T A 2 q T ⋯ A p q T ] , {\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

A special form of matrix [transpose](/source/Transpose) can also be defined for block matrices, where individual blocks are reordered but not transposed. Let A = ( B i j ) {\displaystyle A=(B_{ij})} be a k × l {\displaystyle k\times l} block matrix with m × n {\displaystyle m\times n} blocks B i j {\displaystyle B_{ij}} , the block transpose of A {\displaystyle A} is the l × k {\displaystyle l\times k} block matrix A B {\displaystyle A^{\mathcal {B}}} with m × n {\displaystyle m\times n} blocks ( A B ) i j = B j i {\displaystyle \left(A^{\mathcal {B}}\right)_{ij}=B_{ji}} .[11] As with the conventional trace operator, the block transpose is a [linear mapping](/source/Linear_mapping) such that ( A + C ) B = A B + C B {\displaystyle (A+C)^{\mathcal {B}}=A^{\mathcal {B}}+C^{\mathcal {B}}} .[10] However, in general the property ( A C ) B = C B A B {\displaystyle (AC)^{\mathcal {B}}=C^{\mathcal {B}}A^{\mathcal {B}}} does not hold unless the blocks of A {\displaystyle A} and C {\displaystyle C} commute.

### Addition

Let

B = [ B 11 B 12 ⋯ B 1 s B 21 B 22 ⋯ B 2 s ⋮ ⋮ ⋱ ⋮ B r 1 B r 2 ⋯ B r s ] , {\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}},}

where B i j ∈ C m i × n j {\displaystyle B_{ij}\in \mathbb {C} ^{m_{i}\times n_{j}}} , and let A {\displaystyle A} be the matrix defined in [§ Transpose](#Transpose). (This matrix B {\displaystyle B} will be reused in [§ Multiplication](#Multiplication).) Then if p = r {\displaystyle p=r} , q = s {\displaystyle q=s} , k i = m i {\displaystyle k_{i}=m_{i}} , and ℓ j = n j {\displaystyle \ell _{j}=n_{j}} , then[9]

A + B = [ A 11 + B 11 A 12 + B 12 ⋯ A 1 q + B 1 q A 21 + B 21 A 22 + B 22 ⋯ A 2 q + B 2 q ⋮ ⋮ ⋱ ⋮ A p 1 + B p 1 A p 2 + B p 2 ⋯ A p q + B p q ] . {\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

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](/source/Conformable_matrix) partitions"[12] between two matrices A {\displaystyle A} and B {\displaystyle B} such that all submatrix products that will be used are defined.[13]

Two matrices A {\displaystyle A} and B {\displaystyle B} are said to be partitioned conformally for the product A B {\displaystyle AB} , when A {\displaystyle A} and B {\displaystyle B} are partitioned into submatrices and if the multiplication A B {\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]

Let A {\displaystyle A} be the matrix defined in [§ Transpose](#Transpose), and let B {\displaystyle B} be the matrix defined in [§ Addition](#Addition). Then the matrix product

C = A B {\displaystyle C=AB}

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

C i j = ∑ k = 1 q A i k B k j . {\displaystyle C_{ij}=\sum _{k=1}^{q}A_{ik}B_{kj}.}

Or, using the [Einstein notation](/source/Einstein_notation) that implicitly sums over repeated indices:

C i j = A i k B k j . {\displaystyle C_{ij}=A_{ik}B_{kj}.}

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

C = A B = [ ∑ i = 1 q A 1 i B i 1 ∑ i = 1 q A 1 i B i 2 ⋯ ∑ i = 1 q A 1 i B i s ∑ i = 1 q A 2 i B i 1 ∑ i = 1 q A 2 i B i 2 ⋯ ∑ i = 1 q A 2 i B i s ⋮ ⋮ ⋱ ⋮ ∑ i = 1 q A p i B i 1 ∑ i = 1 q A p i B i 2 ⋯ ∑ i = 1 q A p i B i s ] . {\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

For more details and derivation using block LDU decomposition, see [Schur complement](/source/Schur_complement).

See also: [Helmert–Wolf blocking](/source/Helmert%E2%80%93Wolf_blocking)

If a matrix is partitioned into four blocks, it can be [inverted blockwise](/source/Invertible_matrix#Blockwise_inversion) as follows:

P = [ A B C D ] − 1 = [ A − 1 + A − 1 B ( D − C A − 1 B ) − 1 C A − 1 − A − 1 B ( D − C A − 1 B ) − 1 − ( D − C A − 1 B ) − 1 C A − 1 ( D − C A − 1 B ) − 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}}}

where **A** and **D** are square blocks of arbitrary size, and **B** and **C** are [conformable](/source/Conformable_matrix) with them for partitioning. Furthermore, **A** and the Schur complement of **A** in **P**: **P**/**A** = **D** − **CA**−1**B** must be invertible.[15]

Equivalently, by permuting the blocks:[16]

P = [ A B C D ] − 1 = [ ( A − B D − 1 C ) − 1 − ( A − B D − 1 C ) − 1 B D − 1 − D − 1 C ( A − B D − 1 C ) − 1 D − 1 + D − 1 C ( A − B D − 1 C ) − 1 B D − 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 of **D** in **P**: **P**/**D** = **A** − **BD**−1**C** must be invertible.

If **A** and **D** are both invertible, then:

[ A B C D ] − 1 = [ ( A − B D − 1 C ) − 1 0 0 ( D − C A − 1 B ) − 1 ] [ I − B D − 1 − C A − 1 I ] . {\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 the [Weinstein–Aronszajn identity](/source/Weinstein%E2%80%93Aronszajn_identity), one of the two matrices in the block-diagonal matrix is invertible exactly when the other is. Block matrix inversion also enables to yield from the efficiency of the [fast matrix multiplication algorithms](/source/Computational_complexity_of_matrix_multiplication), which allows to perform the inversion in time O ( n ω ) {\displaystyle O({n^{\omega }})} for 2.37 ≤ ω < 3 {\displaystyle ~2.37\leq \omega <3} [17], Sect. 11, pp. 413-414 .

#### Computing submatrix inverses from the full inverse

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

P = [ A B C D ] , P − 1 = [ E F G H ] {\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 of **P**−1:

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

This relationship follows from recognizing that **E**−1 = **A** − **BD**−1**C** (the Schur complement), and applying the same block inversion formula with the roles of **P** and **P**−1 reversed.[18][19]

### Determinant

The formula for the determinant of a 2 × 2 {\displaystyle 2\times 2} -matrix above continues to hold, under appropriate further assumptions, for a matrix composed of four submatrices A , B , C , D {\displaystyle A,B,C,D} with A {\displaystyle A} and D {\displaystyle D} square. The easiest such formula, which can be proven using either the [Leibniz formula](/source/Leibniz_formula_for_determinants) or a factorization involving the [Schur complement](/source/Schur_complement), is[16] det [ A 0 C D ] = det ( A ) det ( D ) = det [ A B 0 D ] . {\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 that [characteristic polynomials](/source/Characteristic_polynomial) of [ A 0 C D ] {\displaystyle {\begin{bmatrix}A&0\\C&D\end{bmatrix}}} and [ A B 0 D ] {\displaystyle {\begin{bmatrix}A&B\\0&D\end{bmatrix}}} are same and equal to the product of characteristic polynomials of A {\displaystyle A} and D {\displaystyle D} . Furthermore, If [ A 0 C D ] {\displaystyle {\begin{bmatrix}A&0\\C&D\end{bmatrix}}} or [ A B 0 D ] {\displaystyle {\begin{bmatrix}A&B\\0&D\end{bmatrix}}} is [diagonalizable](/source/Diagonalizable), then A {\displaystyle A} and D {\displaystyle D} are diagonalizable too. The converse is false; simply check [ 1 1 0 1 ] {\displaystyle {\begin{bmatrix}1&1\\0&1\end{bmatrix}}} .

If A {\displaystyle A} is [invertible](/source/Invertible_matrix), one has[16]

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

and if D {\displaystyle D} is invertible, one has[20][16]

det [ A B C D ] = det ( D ) det ( A − B D − 1 C ) . {\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 the *same* size further formulas hold. For example, if C {\displaystyle C} and D {\displaystyle D} [commute](/source/Commutativity) (i.e., C D = D C {\displaystyle CD=DC} ), then[21] det [ A B C D ] = det ( A D − B C ) . {\displaystyle \det {\begin{bmatrix}A&B\\C&D\end{bmatrix}}=\det(AD-BC).} Similar statements hold when A B = B A {\displaystyle AB=BA} , A C = C A {\displaystyle AC=CA} , or ⁠ B D = D B {\displaystyle BD=DB} ⁠. Namely, if A C = C A {\displaystyle AC=CA} , then det [ A B C D ] = det ( A D − C B ) . {\displaystyle \det {\begin{bmatrix}A&B\\C&D\end{bmatrix}}=\det(AD-CB).} Note the change in order of C {\displaystyle C} and B {\displaystyle B} (we have C B {\displaystyle CB} instead of B C {\displaystyle BC} ). Similarly, if B D = D B {\displaystyle BD=DB} , then A D {\displaystyle AD} should be replaced with D A {\displaystyle DA} (i.e. we get det ( D A − B C ) {\displaystyle \det(DA-BC)} ) and if A B = B A {\displaystyle AB=BA} , then we should have det ( D A − C B ) {\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 than 2 × 2 {\displaystyle 2\times 2} blocks, again under appropriate commutativity conditions among the individual blocks.[22]

For A = D {\displaystyle A=D} and B = C {\displaystyle B=C} , the following formula holds (even if A {\displaystyle A} and B {\displaystyle B} do not commute)[16] det [ A B B A ] = 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

### Direct sums and block diagonal matrices

#### Direct sum

See also: [Direct sum of matrices](/source/Direct_sum_of_matrices)

For any arbitrary matrices **A** (of size *m* × *n*) and **B** (of size *p* × *q*), we have the **direct sum** of **A** and **B**, denoted by **A** ⊕ **B** and defined as[10]

A ⊕ B = [ a 11 ⋯ a 1 n 0 ⋯ 0 ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ a m 1 ⋯ a m n 0 ⋯ 0 0 ⋯ 0 b 11 ⋯ b 1 q ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ 0 ⋯ 0 b p 1 ⋯ b p q ] . {\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,

[ 1 3 2 2 3 1 ] ⊕ [ 1 6 0 1 ] = [ 1 3 2 0 0 2 3 1 0 0 0 0 0 1 6 0 0 0 0 1 ] . {\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 that **A** and **B** have the same number of dimensions).

Note that any element in the [direct sum](/source/Direct_sum_of_vector_spaces) of two [vector spaces](/source/Vector_space) of matrices could be represented as a direct sum of two matrices.

#### Block diagonal matrices

See also: [Diagonal matrix](/source/Diagonal_matrix)

A **block diagonal matrix** is a block matrix that is a [square matrix](/source/Square_matrix) such that the main-diagonal blocks are square matrices and all off-diagonal blocks are zero matrices.[16] That is, a block diagonal matrix **A** has the form

A = [ A 1 0 ⋯ 0 0 A 2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ A n ] {\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}}}

where **A***k* is a square matrix for all *k* = 1, ..., *n*. In other words, matrix **A** is the [direct sum](/source/Direct_sum_of_matrices) of **A**1, ..., **A***n*.[16] It can also be indicated as **A**1 ⊕ **A**2 ⊕ ... ⊕ **A***n*[10] or diag(**A**1, **A**2, ..., **A***n*)[10] (the latter being the same formalism used for a [diagonal matrix](/source/Diagonal_matrix)). Any square matrix can trivially be considered a block diagonal matrix with only one block.

For the [determinant](/source/Determinant) and [trace](/source/Trace_(linear_algebra)), the following properties hold:

- det A = det A 1 × ⋯ × det A n , {\displaystyle {\begin{aligned}\det {A}&=\det {A}_{1}\times \cdots \times \det {A}_{n},\end{aligned}}} [23][24] and

- tr ⁡ A = tr ⁡ A 1 + ⋯ + tr ⁡ A n . {\displaystyle {\begin{aligned}\operatorname {tr} {A}&=\operatorname {tr} {A}_{1}+\cdots +\operatorname {tr} {A}_{n}.\end{aligned}}} [16][24]

A block diagonal matrix is invertible [if and only if](/source/If_and_only_if) each of its main-diagonal blocks are invertible, and in this case its inverse is another block diagonal matrix given by[25] [ A 1 0 ⋯ 0 0 A 2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ A n ] − 1 = [ A 1 − 1 0 ⋯ 0 0 A 2 − 1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ A n − 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}}.}

The [eigenvalues](/source/Eigenvalues_and_eigenvectors)[26] [and eigenvectors](/source/Eigenvalues_and_eigenvectors) of A {\displaystyle {A}} are simply those of the A k {\displaystyle {A}_{k}} s combined.[24]

### Block tridiagonal matrices

See also: [Tridiagonal matrix](/source/Tridiagonal_matrix)

A **block tridiagonal matrix** is another special block matrix, which is just like the block diagonal matrix a [square matrix](/source/Square_matrix), having square matrices (blocks) in the lower diagonal, [main diagonal](/source/Main_diagonal) and upper diagonal, with all other blocks being zero matrices. It is essentially a [tridiagonal matrix](/source/Tridiagonal_matrix) but has submatrices in places of scalars. A block tridiagonal matrix A {\displaystyle A} has the form

A = [ B 1 C 1 ⋯ 0 A 2 B 2 C 2 ⋱ ⋱ ⋱ ⋮ A k B k C k ⋮ ⋱ ⋱ ⋱ A n − 1 B n − 1 C n − 1 0 ⋯ A n B n ] {\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}}}

where A k {\displaystyle {A}_{k}} , B k {\displaystyle {B}_{k}} and C k {\displaystyle {C}_{k}} are square sub-matrices of the lower, main and upper diagonal respectively.[27][28]

Block tridiagonal matrices are often encountered in numerical solutions of engineering problems (e.g., [computational fluid dynamics](/source/Computational_fluid_dynamics)). Optimized numerical methods for [LU factorization](/source/LU_factorization) are available[29] and hence efficient solution algorithms for equation systems with a block tridiagonal matrix as coefficient matrix. The [Thomas algorithm](/source/Thomas_algorithm), used for efficient solution of equation systems involving a [tridiagonal matrix](/source/Tridiagonal_matrix) can also be applied using matrix operations to block tridiagonal matrices (see also [Block LU decomposition](/source/Block_LU_decomposition)).

### Block triangular matrices

See also: [Triangular matrix](/source/Triangular_matrix)

An n × n {\displaystyle n\times n} matrix A {\displaystyle A} is **upper block triangular** (or **block upper triangular**[30]) if there are positive integers n 1 , … , n k {\displaystyle n_{1},\ldots ,n_{k}} such that n = n 1 + n 2 + … + n k {\displaystyle n=n_{1}+n_{2}+\ldots +n_{k}} and A = [ A 11 A 12 ⋯ A 1 k 0 A 22 ⋯ A 2 k ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ A k k ] , {\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 matrix A i j {\displaystyle A_{ij}} is n i × n j {\displaystyle n_{i}\times n_{j}} for all i , j = 1 , … , k {\displaystyle i,j=1,\ldots ,k} .[26][30] Similarly, A {\displaystyle A} is **lower block triangular** if A = [ A 11 0 ⋯ 0 A 21 A 22 ⋯ 0 ⋮ ⋮ ⋱ ⋮ A k 1 A k 2 ⋯ A k k ] , {\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}},} where A i j {\displaystyle A_{ij}} is n i × n j {\displaystyle n_{i}\times n_{j}} for all i , j = 1 , … , k {\displaystyle i,j=1,\ldots ,k} .[26]

### Block Toeplitz matrices

See also: [Toeplitz matrix](/source/Toeplitz_matrix)

A **block Toeplitz matrix** is another special block matrix, which contains blocks that are repeated down the diagonals of the matrix, as a [Toeplitz matrix](/source/Toeplitz_matrix) has elements repeated down the diagonal.

A matrix A {\displaystyle A} is **block Toeplitz** if A ( i , j ) = A ( k , l ) {\displaystyle A_{(i,j)}=A_{(k,l)}} for all k − i = l − j {\displaystyle k-i=l-j} , that is,

A = [ A 1 A 2 A 3 ⋯ A 4 A 1 A 2 ⋯ A 5 A 4 A 1 ⋯ ⋮ ⋮ ⋮ ⋱ ] , {\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}},}

where A i ∈ F n i × m i {\displaystyle A_{i}\in \mathbb {F} ^{n_{i}\times m_{i}}} .[26]

### Block Hankel matrices

See also: [Hankel matrix](/source/Hankel_matrix)

A matrix A {\displaystyle A} is **block Hankel** if A ( i , j ) = A ( k , l ) {\displaystyle A_{(i,j)}=A_{(k,l)}} for all i + j = k + l {\displaystyle i+j=k+l} , that is,

A = [ A 1 A 2 A 3 ⋯ A 2 A 3 A 4 ⋯ A 3 A 4 A 5 ⋯ ⋮ ⋮ ⋮ ⋱ ] , {\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}},}

where A i ∈ F n i × m i {\displaystyle A_{i}\in \mathbb {F} ^{n_{i}\times m_{i}}} .[26]

## See also

- [Kronecker product](/source/Kronecker_product) (matrix direct product resulting in a block matrix)

- [Jordan normal form](/source/Jordan_normal_form) (canonical form of a linear operator on a finite-dimensional complex vector space)

- [Strassen algorithm](/source/Strassen_algorithm) (algorithm for matrix multiplication that is faster than the conventional matrix multiplication algorithm)

## Notes

1. **[^](#cite_ref-1)** [Eves, Howard](/source/Howard_Eves) (1980). [*Elementary Matrix Theory*](https://archive.org/details/elementarymatrix0000eves_r2m2) (reprint ed.). New York: Dover. p. [37](https://archive.org/details/elementarymatrix0000eves_r2m2/page/37). [ISBN](/source/ISBN_(identifier)) [0-486-63946-0](https://en.wikipedia.org/wiki/Special:BookSources/0-486-63946-0). Retrieved 24 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-called *partitioned*, or *block*, *matrices*.

1. ^ [***a***](#cite_ref-:8_2-0) [***b***](#cite_ref-:8_2-1) Dobrushkin, Vladimir. ["Partition Matrices"](https://www.cfm.brown.edu/people/dobrush/cs52/Mathematica/Part2/partition.html). *Linear Algebra with Mathematica*. Retrieved 2024-03-24.

1. **[^](#cite_ref-3)** Anton, Howard (1994). *Elementary Linear Algebra* (7th ed.). New York: John Wiley. p. 30. [ISBN](/source/ISBN_(identifier)) [0-471-58742-7](https://en.wikipedia.org/wiki/Special:BookSources/0-471-58742-7). A matrix can be subdivided or ***partitioned*** into smaller matrices by inserting horizontal and vertical rules between selected rows and columns.

1. **[^](#cite_ref-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"](http://research.ijcaonline.org/volume93/number6/pxc3895662.pdf) (PDF). *International Journal of Computer Applications*. **93** (6): 13. [Bibcode](/source/Bibcode_(identifier)):[2014IJCA...93f..11I](https://ui.adsabs.harvard.edu/abs/2014IJCA...93f..11I). [doi](/source/Doi_(identifier)):[10.5120/16218-5662](https://doi.org/10.5120%2F16218-5662).

1. **[^](#cite_ref-5)** Macedo, H.D.; Oliveira, J.N. (2013). "Typing linear algebra: A biproduct-oriented approach". *Science of Computer Programming*. **78** (11): 2160–2191. [arXiv](/source/ArXiv_(identifier)):[1312.4818](https://arxiv.org/abs/1312.4818). [doi](/source/Doi_(identifier)):[10.1016/j.scico.2012.07.012](https://doi.org/10.1016%2Fj.scico.2012.07.012).

1. ^ [***a***](#cite_ref-:3_6-0) [***b***](#cite_ref-:3_6-1) [***c***](#cite_ref-:3_6-2) Johnston, Nathaniel (2021). *Introduction to linear and matrix algebra*. Cham, Switzerland: Springer Nature. pp. 30, 425. [ISBN](/source/ISBN_(identifier)) [978-3-030-52811-9](https://en.wikipedia.org/wiki/Special:BookSources/978-3-030-52811-9).

1. ^ [***a***](#cite_ref-:4_7-0) [***b***](#cite_ref-:4_7-1) Johnston, Nathaniel (2021). *Advanced linear and matrix algebra*. Cham, Switzerland: Springer Nature. p. 298. [ISBN](/source/ISBN_(identifier)) [978-3-030-52814-0](https://en.wikipedia.org/wiki/Special:BookSources/978-3-030-52814-0).

1. **[^](#cite_ref-8)** Jeffrey, Alan (2010). [*Matrix operations for engineers and scientists: an essential guide in linear algebra*](https://www.worldcat.org/title/639165077). Dordrecht [Netherlands]; New York: Springer. p. 54. [ISBN](/source/ISBN_(identifier)) [978-90-481-9273-1](https://en.wikipedia.org/wiki/Special:BookSources/978-90-481-9273-1). [OCLC](/source/OCLC_(identifier)) [639165077](https://search.worldcat.org/oclc/639165077).

1. ^ [***a***](#cite_ref-:2_9-0) [***b***](#cite_ref-:2_9-1) [***c***](#cite_ref-:2_9-2) [***d***](#cite_ref-:2_9-3) [***e***](#cite_ref-:2_9-4) [***f***](#cite_ref-:2_9-5) [***g***](#cite_ref-:2_9-6) [***h***](#cite_ref-:2_9-7) [***i***](#cite_ref-:2_9-8) [***j***](#cite_ref-:2_9-9) [***k***](#cite_ref-:2_9-10) [***l***](#cite_ref-:2_9-11) [***m***](#cite_ref-:2_9-12) Stewart, Gilbert W. (1998). *Matrix algorithms. 1: Basic decompositions*. Philadelphia, PA: Soc. for Industrial and Applied Mathematics. pp. 18–20. [ISBN](/source/ISBN_(identifier)) [978-0-89871-414-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-89871-414-2).

1. ^ [***a***](#cite_ref-:1_10-0) [***b***](#cite_ref-:1_10-1) [***c***](#cite_ref-:1_10-2) [***d***](#cite_ref-:1_10-3) [***e***](#cite_ref-:1_10-4) Gentle, 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](/source/ISBN_(identifier)) [978-0-387-70873-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-70873-7).

1. **[^](#cite_ref-11)** Mackey, D. Steven (2006). [*Structured linearizations for matrix polynomials*](http://eprints.maths.manchester.ac.uk/314/1/mackey06.pdf) (PDF) (Thesis). University of Manchester. [ISSN](/source/ISSN_(identifier)) [1749-9097](https://search.worldcat.org/issn/1749-9097). [OCLC](/source/OCLC_(identifier)) [930686781](https://search.worldcat.org/oclc/930686781).

1. **[^](#cite_ref-12)** [Eves, Howard](/source/Howard_Eves) (1980). [*Elementary Matrix Theory*](https://archive.org/details/elementarymatrix0000eves_r2m2) (reprint ed.). New York: Dover. p. [37](https://archive.org/details/elementarymatrix0000eves_r2m2/page/37). [ISBN](/source/ISBN_(identifier)) [0-486-63946-0](https://en.wikipedia.org/wiki/Special:BookSources/0-486-63946-0). Retrieved 24 April 2013. A partitioning as in Theorem 1.9.4 is called a *conformable partition* of *A* and *B*.

1. **[^](#cite_ref-13)** Anton, Howard (1994). *Elementary Linear Algebra* (7th ed.). New York: John Wiley. p. 36. [ISBN](/source/ISBN_(identifier)) [0-471-58742-7](https://en.wikipedia.org/wiki/Special:BookSources/0-471-58742-7). ...provided the sizes of the submatrices of A and B are such that the indicated operations can be performed.

1. **[^](#cite_ref-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](/source/ISBN_(identifier)) [978-3-11-056259-0](https://en.wikipedia.org/wiki/Special:BookSources/978-3-11-056259-0).

1. **[^](#cite_ref-15)** Bernstein, Dennis (2005). *Matrix Mathematics*. Princeton University Press. p. 44. [ISBN](/source/ISBN_(identifier)) [0-691-11802-7](https://en.wikipedia.org/wiki/Special:BookSources/0-691-11802-7).

1. ^ [***a***](#cite_ref-:0_16-0) [***b***](#cite_ref-:0_16-1) [***c***](#cite_ref-:0_16-2) [***d***](#cite_ref-:0_16-3) [***e***](#cite_ref-:0_16-4) [***f***](#cite_ref-:0_16-5) [***g***](#cite_ref-:0_16-6) [***h***](#cite_ref-:0_16-7) Abadir, Karim M.; Magnus, Jan R. (2005). *Matrix Algebra*. Cambridge University Press. pp. 97, 100, 106, 111, 114, 118. [ISBN](/source/ISBN_(identifier)) [9781139443647](https://en.wikipedia.org/wiki/Special:BookSources/9781139443647).

1. **[^](#cite_ref-17)** Pan, V. (1984). "How can we speed up matrix multiplication?". *SIAM Review*. **26** (3): 393–415. [doi](/source/Doi_(identifier)):[10.1137/1026076](https://doi.org/10.1137%2F1026076).

1. **[^](#cite_ref-18)** ["Is this formula for a matrix block inverse in terms of the entire matrix inverse known?"](https://mathoverflow.net/questions/495299/is-this-formula-for-a-matrix-block-inverse-in-terms-of-the-entire-matrix-inverse). *MathOverflow*.

1. **[^](#cite_ref-19)** Escalante-B., Alberto N.; Wiskott, Laurenz (2016). ["Improved graph-based SFA: Information preservation complements the slowness principle"](https://www.scielo.org.mx/scielo.php?script=sci_arttext&pid=S1405-55462016000200251). *Machine Learning*. [arXiv](/source/ArXiv_(identifier)):[1412.4679](https://arxiv.org/abs/1412.4679). [doi](/source/Doi_(identifier)):[10.1007/s10994-016-5563-y](https://doi.org/10.1007%2Fs10994-016-5563-y).

1. **[^](#cite_ref-20)** Taboga, Marco (2021). "Determinant of a block matrix", Lectures on matrix algebra.

1. **[^](#cite_ref-21)** Silvester, J. R. (2000). ["Determinants of Block Matrices"](https://web.archive.org/web/20150318222335/http://www.ee.iisc.ernet.in/new/people/faculty/prasantg/downloads/blocks.pdf) (PDF). *Math. Gaz*. **84** (501): 460–467. [doi](/source/Doi_(identifier)):[10.2307/3620776](https://doi.org/10.2307%2F3620776). [JSTOR](/source/JSTOR_(identifier)) [3620776](https://www.jstor.org/stable/3620776). Archived from [the original](http://www.ee.iisc.ernet.in/new/people/faculty/prasantg/downloads/blocks.pdf) (PDF) on 2015-03-18. Retrieved 2021-06-25.

1. **[^](#cite_ref-22)** Sothanaphan, Nat (January 2017). "Determinants of block matrices with noncommuting blocks". *Linear Algebra and Its Applications*. **512**: 202–218. [arXiv](/source/ArXiv_(identifier)):[1805.06027](https://arxiv.org/abs/1805.06027). [doi](/source/Doi_(identifier)):[10.1016/j.laa.2016.10.004](https://doi.org/10.1016%2Fj.laa.2016.10.004). [S2CID](/source/S2CID_(identifier)) [119272194](https://api.semanticscholar.org/CorpusID:119272194).

1. **[^](#cite_ref-23)** Quarteroni, Alfio; Sacco, Riccardo; Saleri, Fausto (2000). *Numerical mathematics*. Texts in applied mathematics. New York: Springer. pp. 10, 13. [ISBN](/source/ISBN_(identifier)) [978-0-387-98959-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-98959-4).

1. ^ [***a***](#cite_ref-:6_24-0) [***b***](#cite_ref-:6_24-1) [***c***](#cite_ref-:6_24-2) George, Raju K.; Ajayakumar, Abhijith (2024). ["A Course in Linear Algebra"](https://doi.org/10.1007/978-981-99-8680-4). *University Texts in the Mathematical Sciences*: 35, 407. [doi](/source/Doi_(identifier)):[10.1007/978-981-99-8680-4](https://doi.org/10.1007%2F978-981-99-8680-4). [ISBN](/source/ISBN_(identifier)) [978-981-99-8679-8](https://en.wikipedia.org/wiki/Special:BookSources/978-981-99-8679-8). [ISSN](/source/ISSN_(identifier)) [2731-9318](https://search.worldcat.org/issn/2731-9318).

1. **[^](#cite_ref-25)** Prince, Simon J. D. (2012). *Computer vision: models, learning, and inference*. New York: Cambridge university press. p. 531. [ISBN](/source/ISBN_(identifier)) [978-1-107-01179-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-107-01179-3).

1. ^ [***a***](#cite_ref-:5_26-0) [***b***](#cite_ref-:5_26-1) [***c***](#cite_ref-:5_26-2) [***d***](#cite_ref-:5_26-3) [***e***](#cite_ref-:5_26-4) Bernstein, Dennis S. (2009). *Matrix mathematics: theory, facts, and formulas* (2 ed.). Princeton, NJ: Princeton University Press. pp. 168, 298. [ISBN](/source/ISBN_(identifier)) [978-0-691-14039-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-14039-1).

1. **[^](#cite_ref-27)** Dietl, Guido K. E. (2007). [*Linear estimation and detection in Krylov subspaces*](https://www.worldcat.org/title/ocm85898525). Foundations in signal processing, communications and networking. Berlin; New York: Springer. pp. 85, 87. [ISBN](/source/ISBN_(identifier)) [978-3-540-68478-7](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-68478-7). [OCLC](/source/OCLC_(identifier)) [85898525](https://search.worldcat.org/oclc/85898525).

1. **[^](#cite_ref-28)** Horn, Roger A.; Johnson, Charles R. (2017). *Matrix analysis* (Second edition, corrected reprint ed.). New York, NY: Cambridge University Press. p. 36. [ISBN](/source/ISBN_(identifier)) [978-0-521-83940-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-83940-2).

1. **[^](#cite_ref-29)** Datta, Biswa Nath (2010). *Numerical linear algebra and applications* (2 ed.). Philadelphia, Pa: SIAM. p. 168. [ISBN](/source/ISBN_(identifier)) [978-0-89871-685-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-89871-685-6).

1. ^ [***a***](#cite_ref-:7_30-0) [***b***](#cite_ref-:7_30-1) Stewart, Gilbert W. (2001). *Matrix algorithms. 2: Eigensystems*. Philadelphia, Pa: Soc. for Industrial and Applied Mathematics. p. 5. [ISBN](/source/ISBN_(identifier)) [978-0-89871-503-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-89871-503-3).

## References

- [Strang, Gilbert](/source/Gilbert_Strang) (1999). ["Lecture 3: Multiplication and inverse matrices"](http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-3-multiplication-and-inverse-matrices). MIT Open Course ware. 18:30–21:10.

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