Block Diagonal Matrix Vector Multiplication

02 Jun, 2021

Given a matrix block diagonal matrix as follows. Viewed 2 times 0 I have to compute many matrix products of matrices that are block-diagonal in a minimisation procedure.

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Let A2IRm n and x2IRn.

Block diagonal matrix vector multiplication. X 2 100. The term x is called the jth component of x. A full matrix if neither nblocks nor blocksizes given ow.

In particular I want to speed up two operations. In terms of its coordi-nates or components we can also write x 2 6 6 4 x 1 x 2. Do i1n x i ii10D00 do j1n A ij ijj100D00 end do end do.

In terms of its coordinates or components we can also write x 2 6 6 6 4 x 1 x 2. P A D a 1 0 0 0 0 a 2 0 0 0 0 a 3 0 0 0 0 a 4 d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d 9 d 10 d 11 d 12 d 13 d 14 d 15 d 16. Zj Aej a1j a2j amj a1j 1 0 0 a2j 0 1 0 amj 0 0 1 m i 1aijˆei where we have written the standard basis of Rm as ˆe1 ˆe2 ˆem.

Block multiplication has theoretical uses as we shall see. In doing exercise 1610 in Linear Algebra and Its Applications I was reminded of the general issue of multiplying block matrices including diagonal block matrices. Length X needs to be a multiple of dim mat 1.

It defaults to length blocksizes. Let M be a square matrix having a triangular. The matrices are partitioned into blocks in such a way that each product of blocks can be handled.

X 11111 tempMatrix. Then the matrix M 2 is block diagonal having M 2 jj along the diagonal and all powers M k are block diagonal having Mk jj along the diagonal. A 11 2 4 1 2 3 0 4 5 0 0 6 3 5.

A 22 7 0 8 9. I have to generate a matrix using. We de ne the matrix-vector product Axby Ax 2 6 6 6 6 6 4 A 1 x A.

X 3 22. Has eigenvalues 1 4 6 7 and 9. That is use the outer product with another vector which contains only 1 entries.

26 Sparse Matrix-Vector Multiplication Mechanism. Number of diagonal blocks in mat or R. Then the blocks are stored in auxiliary memory and their products are computed one by one.

X 1 5. REVIEW OF MATRICES AND BLOCK STRUCTURES 11. Ask Question Asked today.

Block Sparse Matrix-Vector Multiplication with CUDA. For Lapack I simply use CALL DGEMV Nnn10D00anx100D00b1 How do I achieve this in PBLAS. X 2 100.

However we cannot make the matrix sizes arbitrarily large because all three blocks have to fit inside the memory. I then discussed block diagonal matrices ie block matrices in which the off-diagonal submatrices are zero and in a multipart series of posts showed that we can uniquely and maximally partition any square matrix into block. In the previous post weve discussed sparse matrix-vector multiplication.

Is it correct to say that the product of this matrix and any other matrix for example D. Now apply the hadamard multiplication to this tempMatrix with the identity matrix. For example if x 2 4 5 100 22 3 5.

It was shown that its possible to. X 3 22. If fast memory has size M fast 3b 2 M fast q b M fast 3 12.

X 1 5. To get half of the machine peak capacity q t m t f Therefore to run blocked matrix multiplication at half of the peak machine capacity 3q 2 3b 2. In a previous post I discussed the general problem of multiplying block matrices ie matrices partitioned into multiple submatrices.

A block diagonal square matrix. The term x j is called the jth component of x. Here is what I must do.

However it is also useful in computing products of matrices in a computer with limited memory capacity. And then simply multiply Matrix A by Vector x bAx. 30 28 Storage Space for Increasing Number of Non-zeros in Matrix.

Now you have a comprehensive understanding of blocked matrix multiplication. X n 3 7 7 5with each x j 2IR. A 1 0.

Determinant of block matrices. First define a grouping vector g which has a unique value for each group. We de ne the matrix-vector product Axby.

On order to do so you need first to build a matrix out of the vector x. These are called the row and column block representations of A respectively 11. Furthermore any linear combination of these powers of Msayc 1Mc 2M 2 is block diagonal having the corresponding c 1Mjj c 2Mjj 2 along the diagonal.

26 27 Example of the Hierarchical Sparse Matrix Storage Format. Defaults to 1 ie. Vector or matrix to multiply by mat.

In other words if zj Aej is written as a linear combination of the basis vectors of Rm the element aij is the coefficient corresponding to ˆei. Then use the formula shown which creates an n x n block diagonal matrix with elements of a having the same dimensions as a and multiplies that by an n x ng block diagonal matrix with the elements of. For example if x 2 4 5 100 22 3 5.

This also came up in exercise 1424 as well which I answered without necessarily fully understanding the problem. X n 3 7 7 7 5 with each x j2R. This is because Ahas a block upper-triangular structure A A 11 A 12 0 A 22.

A a 1 0 0 0 0 a 2 0 0 0 0 a 3 0 0 0 0 a 4 where each non zero entry a i is M 1 vector and the zero are also M 1 vectors. I therefore wanted to go back and try to prove various claims about block matrices starting with the following. 21 hours agoFast numpy multiplication of block diagonal matrix with normal matrix.

Because both of these blocks are themselves triangular their eigenvalues are equal to their diagonal elements and the spectrum of Ais the union of the spectra of these blocks. Let A2Rm n and x2Rn.

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