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Chapter 7: Problem 3

Let \(A\) and \(B\) be \(n \times n\) matrices and let \(\mathbf{x} \in \mathbb{R}^{n}\) (a) How many scalar additions and multiplications are necessary to compute the product \(A \mathbf{x} ?\) (b) How many scalar additions and multiplications are necessary to compute the product \(A B ?\) (c) How many scalar additions and multiplications are necessary to compute \((A B) \mathbf{x} ?\) To compute \(A(B \mathbf{x}) ?\)

Short Answer

Expert verified
(a) To compute the product \(A \mathbf{x}\), we need \(n^2\) multiplications and \((n - 1)n\) additions. (b) To compute the product \(A B\), we need \(n^3\) multiplications and \((n - 1)n^2\) additions. (c) To compute the products (AB)\(\mathbf{x}\) and A(B\(\mathbf{x}\)): - For (AB)\(\mathbf{x}\), we need \(n^3 + n^2\) multiplications and \((n - 1)n^2 + (n - 1)n\) additions. - For A(B\(\mathbf{x}\)), we need \(2n^2\) multiplications and \(2(n - 1)n\) additions.

Step by step solution

01

Part (a): Compute A饾惐 Product

To compute the product of a matrix A and a vector 饾惐, both with n dimensions, we need to perform \(n^2\) multiplications and \((n - 1)n\) additions. Here's why: 1. Matrix A has n rows and n columns, which means each row in A needs to be multiplied with the corresponding element in 饾惐. 2. The result of each row and 饾惐 element multiplication is summed up, contributing to the final result, which is a n-dimensional vector. So for each row, n multiplications and (n - 1) additions are performed. Since there are n rows, we have a total of \(n^2\) multiplications and \((n - 1)n\) additions.
02

Part (b): Compute AB Product

To compute the product of two n x n matrices A and B, we need to perform \(n^3\) multiplications and \((n - 1)n^2\) additions. Here's why: 1. To calculate an element in the resulting matrix (C = AB), the corresponding row from matrix A and column from matrix B need to be used. 2. Multiply and sum the element-wise products of the row and column. This results in n multiplications and (n - 1) additions. 3. These operations are performed for each element in the resulting matrix C, which has \(n^2\) elements. So for each element, n multiplications and (n - 1) additions are made. As there are \(n^2\) elements, we have a total of \(n^3\) multiplications and \((n - 1)n^2\) additions.
03

Part (c): Compute (AB)饾惐 and A(B饾惐) Products

First, we calculate the number of scalar additions and multiplications for the (AB)饾惐 product: 1. Compute the AB product, which involves \(n^3\) multiplications and \((n - 1)n^2\) additions, as seen in Part (b). 2. Multiply the resulting matrix (AB) by the vector 饾惐. This requires \(n^2\) multiplications and \((n - 1)n\) additions, as seen in Part (a). Adding the number of operations from steps 1 and 2, we have a total of \(n^3 + n^2\) multiplications and \((n - 1)n^2 + (n - 1)n\) additions for computing (AB)饾惐. Next, we calculate the number of scalar additions and multiplications for the A(B饾惐) product: 1. Multiply matrix B by vector 饾惐. This requires \(n^2\) multiplications and \((n - 1)n\) additions, as seen in Part (a). 2. Multiply the resulting vector (B饾惐) by matrix A. This requires an additional \(n^2\) multiplications and \((n - 1)n\) additions, also based on Part (a). Adding the number of operations from steps 1 and 2, we have a total of \(2n^2\) multiplications and \(2(n - 1)n\) additions for computing A(B饾惐).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scalar Addition
Scalar addition plays a fundamental role in matrix operations, as it involves the addition of individual elements. While these operations seem simple, they are crucial in computing matrix and vector products. When multiplying a matrix by a vector, like in matrix-vector product scenarios, multiple scalar additions are required to sum the products of row and vector element multiplications.
These sums then contribute to the final entries of the resulting vector or matrix. For example, in the computation of the product \(A \mathbf{x}\), with \(n\) rows, each resulting element of the vector necessitates \((n - 1)\) scalar additions. Matrix multiplication \((AB)\) is even more complex, needing \((n - 1)n^2\) additions to form the matrices鈥� final product. Scalar addition, though often seen as a basic operation, underpins the computational effort involved in matrix arithmetic.
Scalar Multiplication
Scalar multiplication is the process of multiplying matrix elements by values, playing a vital role in matrix computations such as matrix-vector and matrix-matrix products. In matrix-vector multiplication, every element in a row of the matrix is multiplied by a corresponding element in the vector.
This results in \(n^2\) multiplications if both matrices and vectors are of dimension \(n\). When considering the multiplication of two \(n \times n\) matrices, like \(AB\), the number of scalar multiplications surges to \(n^3\). This reflects how computational demands increase dramatically with the size and complexity of the matrix operations being performed.
Matrix-Vector Product
The matrix-vector product is a basic yet crucial concept involving processes where a matrix multiplies a vector. To compute this, every row of the matrix is treated separately with each element being multiplied with the corresponding element of the vector.
This produces a series of products that are then summed up to form the resultant vector. The complexity comes from the number of operations needed: \(n^2\) multiplications and \(n-1)\) additions are necessary for a matrix \(A\) with \(n\) rows and columns, and a vector \(\mathbf{x}\) with \(n\) entries.
  • This operation transforms vectors in application areas like computer graphics, physics simulations, and data transformations.
  • Efficiency in performing these multiplications can significantly affect computational time in larger systems.
Computational Complexity
Computational complexity refers to the efficiency and resource usage of matrix operations like matrix multiplication. It quantifies the number of operations (additions and multiplications) needed to complete a task. This is crucial in performance-focused applications such as scientific computing and large-scale simulations.
As demonstrated, matrix multiplication (\(AB\)) demands \(n^3\) scalar multiplications and \((n - 1)n^2\) additions, highlighting the growing complexity with an increase in matrix size. Understanding this complexity helps software architects and engineers design algorithms and systems that manage resources effectively and select optimal strategies for matrix operations. Efficient algorithms minimize resource input, saving time and computational power on a large scale. As such, a grasp of computational complexity is vital for endeavors in data science, machine learning, and beyond.

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Most popular questions from this chapter

What would the machine epsilon be for a computer that uses 36 -digit base 2 floating-point arithmetic?

Let \(x_{1}=94,210, x_{2}=8631, x_{3}=1440, x_{4}=133\) and \(x_{5}=34 .\) Calculate each of the following, using four-digit decimal floating-point arithmetic: (a) \(\left(\left(\left(x_{1}+x_{2}\right)+x_{3}\right)+x_{4}\right)+x_{5}\) (b) \(x_{1}+\left(\left(x_{2}+x_{3}\right)+\left(x_{4}+x_{5}\right)\right)\) \((\mathbf{c})\left(\left(\left(x_{5}+x_{4}\right)+x_{3}\right)+x_{2}\right)+x_{1}\)

Theorem 7.4 .2 states that \\[ \|A\|_{\infty}=\max _{1 \leq i \leq m}\left(\sum_{j=1}^{n}\left|a_{i j}\right|\right) \\] Prove this in two steps. (a) Show first that \\[ \|A\|_{\infty} \leq \max _{1 \leq i \leq m}\left(\sum_{j=1}^{n}\left|a_{i j}\right|\right) \\] (b) Construct a vector \(\mathbf{x}\) whose coordinates are each 卤1 such that \\[ \frac{\|A \mathbf{x}\|_{\infty}}{\|\mathbf{x}\|_{\infty}}=\|A \mathbf{x}\|_{\infty}=\max _{1 \leq i \leq m}\left(\sum_{j=1}^{n}\left|a_{i j}\right|\right) \\]

Solve the given two systems and compare the solutions. Are the coefficient matrices well conditioned? Ill conditioned? Explain. \\[ \begin{aligned} 1.0 x_{1}+2.0 x_{2} &=1.12 \\ 1.000 x_{1}+2.011 x_{2} &=1.120 \\ 2.0 x_{1}+3.9 x_{2} &=2.16 \\ 2.000 x_{1}+3.982 x_{2} &=2.160 \end{aligned} \\]

\(\operatorname{Let} Q^{T}=G_{n-k} \cdots G_{2} G_{1},\) where each \(G_{i}\) is a Givens transformation. Let \(\mathbf{b} \in \mathbb{R}^{n}\) and let \(A\) be an \(n \times n\) matrix. How many additions and multiplications are necessary to compute (a) \(Q^{T} \mathbf{b} ;\) (b) \(Q^{T} A ?\)
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