Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Block Matrix Multiplication**
*Introduction:*
In linear algebra, matrices can be partitioned into smaller submatrices, which allows for simplification of complex matrix operations. This section illustrates how to multiply two 4x4 matrices, A and B, using their 2x2 block submatrices.
*Block Partitioning of Matrices:*
Consider matrices A and B, both of size 4x4. They are partitioned into 2x2 submatrices or "blocks" as follows:
Matrix \( A \):
\[
A = \begin{bmatrix}
A_{11} & A_{12} \\
A_{21} & A_{22}
\end{bmatrix}
\]
Matrix \( B \):
\[
B = \begin{bmatrix}
B_{11} & B_{12} \\
B_{21} & B_{22}
\end{bmatrix}
\]
*Blockwise Matrix Multiplication:*
The product \( AB \) can be computed using blockwise multiplication. The resulting matrix is:
\[
AB = \begin{bmatrix}
A_{11}B_{11} + A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22} \\
A_{21}B_{11} + A_{22}B_{21} & A_{21}B_{12} + A_{22}B_{22}
\end{bmatrix}
\]
*Explanation:*
- The element at position (1,1) in the product matrix \( AB \) is obtained by multiplying block \( A_{11} \) with \( B_{11} \) and adding the product of \( A_{12} \) with \( B_{21} \).
- Similarly, for the element at position (1,2), multiply \( A_{11} \) with \( B_{12} \) and add it to the product of \( A_{12} \) with \( B_{22} \).
- For the element at position (2,1), calculate the sum of the products of \( A_{21} \) with \( B_{11} \) and \( A_{22} \) with \( B_{21} \).
- Finally, the element at position (2,2) is the sum of \( A_{21}B_{12](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80a13d8c-5188-4010-ba23-710cc489cc25%2F6301e9f4-ac03-403e-80c1-97cf59783139%2Ff1d4o17_processed.png&w=3840&q=75)
Transcribed Image Text:**Block Matrix Multiplication**
*Introduction:*
In linear algebra, matrices can be partitioned into smaller submatrices, which allows for simplification of complex matrix operations. This section illustrates how to multiply two 4x4 matrices, A and B, using their 2x2 block submatrices.
*Block Partitioning of Matrices:*
Consider matrices A and B, both of size 4x4. They are partitioned into 2x2 submatrices or "blocks" as follows:
Matrix \( A \):
\[
A = \begin{bmatrix}
A_{11} & A_{12} \\
A_{21} & A_{22}
\end{bmatrix}
\]
Matrix \( B \):
\[
B = \begin{bmatrix}
B_{11} & B_{12} \\
B_{21} & B_{22}
\end{bmatrix}
\]
*Blockwise Matrix Multiplication:*
The product \( AB \) can be computed using blockwise multiplication. The resulting matrix is:
\[
AB = \begin{bmatrix}
A_{11}B_{11} + A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22} \\
A_{21}B_{11} + A_{22}B_{21} & A_{21}B_{12} + A_{22}B_{22}
\end{bmatrix}
\]
*Explanation:*
- The element at position (1,1) in the product matrix \( AB \) is obtained by multiplying block \( A_{11} \) with \( B_{11} \) and adding the product of \( A_{12} \) with \( B_{21} \).
- Similarly, for the element at position (1,2), multiply \( A_{11} \) with \( B_{12} \) and add it to the product of \( A_{12} \) with \( B_{22} \).
- For the element at position (2,1), calculate the sum of the products of \( A_{21} \) with \( B_{11} \) and \( A_{22} \) with \( B_{21} \).
- Finally, the element at position (2,2) is the sum of \( A_{21}B_{12
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