Problem 45 - Differential Equations and Linear Algebra

Question

Let \(A=\left[a_{i j}(t)\right]\) be an \(m \times n\) matrix function and let \(B=\left[b_{i j}(t)\right]\) be an \(n \times p\) matrix function. Use the definition of matrix multiplication to prove that $$ \frac{d}{d t}(A B)=A \frac{d B}{d t}+\frac{d A}{d t} B $$

Step-by-Step Solution

Answer

To prove \(\frac{d}{dt}(AB) = A \frac{dB}{dt} + \frac{dA}{dt} B\), we define the product matrix \(C = AB\), with elements \(c_{ij}(t) = \sum_{k=1}^n a_{ik}(t) b_{kj}(t)\). Differentiate each element of C with respect to t, and apply the product rule: \(\frac{dc_{ij}(t)}{dt} = \sum_{k=1}^n \left( \frac{da_{ik}(t)}{dt} b_{kj}(t) + a_{ik}(t) \frac{db_{kj}(t)}{dt} \right)\). Rewrite this as a matrix product: \(\frac{dC}{dt} = \frac{dA}{dt}B + A\frac{dB}{dt}\). This shows that the derivative of the product matrix C equals the sum of products of the derivatives of A and B with B and A, respectively, thereby proving the given equation.

Step 1: Define the product matrix C

First, let's denote the product matrix of A and B as C, that is, \(C = AB\), with dimensions m x p. Its elements can be defined as: \[c_{ij}(t) = \sum_{k=1}^n a_{ik}(t) b_{kj}(t)\]

Step 2: Compute the derivative of each element in C

To find the derivative of C with respect to time (t), we need to differentiate each element of the matrix C with respect to t. Thus, we have: \[\frac{dc_{ij}(t)}{dt} = \frac{d}{dt} \left( \sum_{k=1}^n a_{ik}(t) b_{kj}(t) \right)\]

Step 3: Use the product rule for differentiation

Now, using the product rule of differentiation, we can rewrite the derivative as follows: \[\frac{dc_{ij}(t)}{dt} = \sum_{k=1}^n \left( \frac{da_{ik}(t)}{dt} b_{kj}(t) + a_{ik}(t) \frac{db_{kj}(t)}{dt} \right)\]

Step 4: Rewrite the sum in matrix form

We can now rewrite the above expression as a matrix product: \[\frac{dC}{dt} = \frac{dA}{dt}B + A\frac{dB}{dt}\]

Step 5: Conclude the proof

Since we have shown that the derivative of the product matrix C is equal to the sum of the products of the derivatives of A and B with B and A respectively, we have proved the given equation: \[\frac{d}{dt}(AB) = A \frac{d B}{d t}+\frac{d A}{d t} B\]

Key Concepts

Matrix Multiplication

Matrix Multiplication is a fundamental operation in linear algebra where two matrices are combined to form a new matrix. Suppose you have an \(m \times n\) matrix \(A\) and an \(n \times p\) matrix \(B\). The result of the multiplication is a new matrix \(C\) with dimensions \(m \times p\). Here's how we calculate each element of \(C\):

• To find an element \(c_{ij}\) of \(C\), you take the dot product of the \(i^{th}\) row of \(A\) and the \(j^{th}\) column of \(B\).
• This involves multiplying corresponding elements from the row and the column and summing the results: \[c_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}\]

This operation is not commutative, meaning \(AB eq BA\) in general. Understanding matrix multiplication is essential since it lays the groundwork for more complex operations, like differentiation of matrix products.

Product Rule for Differentiation

The Product Rule is a crucial principle used in calculus to differentiate products of functions. In the context of matrices, it allows us to find the derivative of a product of two matrix functions. When you have two differentiable matrix functions, \(A(t)\) and \(B(t)\), the product rule states that:

• The derivative of the product \(AB\) with respect to \(t\) is \[\frac{d}{dt}(AB) = A \frac{dB}{dt} + \frac{dA}{dt} B\]
• This resembles the familiar single-variable product rule: \[\frac{d}{dt}(uv) = u \frac{dv}{dt} + \frac{du}{dt} v\]
• The rule emphasizes that both terms are derivatives and must account for changes in each component separately and together.

Understanding this rule prevents mistakes when working with dynamic systems or problems involving changing variables.

Derivative of Matrices

Taking derivatives of matrices extends the principles of calculus to multidimensional arrays. Each element of a matrix can be seen as a function of a variable, typically time \(t\). To differentiate a matrix, you differentiate each element individually with respect to \(t\):

• For a matrix \(A = [a_{ij}(t)]\), the derivative is another matrix \(\frac{dA}{dt} = \left[ \frac{da_{ij}}{dt} \right]\).
• This process is similar to differentiating scalar functions but applied element-wise to multi-dimensional arrays.
• Derivatives of matrices are useful in many fields, including statistics, physics, and engineering, where systems are modeled using state, output, and input matrices.

Mastering matrix derivatives makes it possible to tackle complex, real-world problems involving numerous variables and their interrelationships.