What is the compatibility of block matrix multiplication?

What is the compatibility of block matrix multiplication?

The only requirement is that the blocks be compatible. That is, the sizes of the blocks must be such that all matrix products of blocks that occur make sense. This means that the number of columns in each block of must equal the number of rows in the corresponding block of .

What is block form of a matrix?

In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. 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.

How do you find the determinant of a block matrix?

det(M)=det(A−BD−1C)det(D). det ( M ) = det ( A − B D − 1 C ) det ( D ) . (the determinant of a block triangular matrix is the product of the determinants of its diagonal blocks).

What is block lower triangular matrix?

Determinant of a block-triangular matrix because we are dealing with a triangular matrix having all the diagonal entries equal to 1. A block-lower-triangular matrix is a matrix of the form where and. are square matrices. Proposition Let be a block-lower-triangular matrix, as defined above.

What is the inverse of a block matrix?

Notice that the inverse of a block diagonal matrix is also block diagonal. Similarly, the inverse of a block secondary diagonal matrix is block secondary diagonal too, but in transposed partition so that there is a switch between B and C.

When can you not multiply matrices?

You can only multiply two matrices if their dimensions are compatible , which means the number of columns in the first matrix is the same as the number of rows in the second matrix.

How do you find the inverse of a matrix using partition?

The partition method is based on the fact that if the inverse of square matrix An of order n is known, then the inverse of the matrix An+1 A n + 1 can be obtained by adding (n+1)th ( n + 1 ) t h row and (n+1)th ( n + 1 ) t h column to An .

How do you determine if matrices can be multiplied?

You can only multiply two matrices if their dimensions are compatible , which means the number of columns in the first matrix is the same as the number of rows in the second matrix. If A=[aij] is an m×n matrix and B=[bij] is an n×p matrix, the product AB is an m×p matrix.

What is a block in math example?

It is often useful to consider matrices whose entries are themselves matrices, called blocks. A matrix viewed in this way is said to be partitioned into blocks B = [ b 1 b 2 … b k] where the b j are the columns of B is such a block partition of B. Here is another example.

How do you find the product of matrices in blocks?

If matrices A and B are partitioned compatibly into blocks, the product A B can be computed by matrix multiplication using blocks as entries. We omit the proof. Block multiplication has theoretical uses as we shall see. However, it is also useful in computing products of matrices in a computer with limited memory capacity.

What is block multiplication in Computer Science?

Block multiplication has theoretical uses as we shall see. However, it is also useful in computing products of matrices in a computer with limited memory capacity. The matrices are partitioned into blocks in such a way that each product of blocks can be handled.

Can we compute the product A B using blocks as entries?

In other words, we can compute the product A B by ordinary matrix multiplication, using blocks as entries. The only requirement is that the blocks be compatible. That is, the sizes of the blocks must be such that all matrix products of blocks that occur make sense.