Problem 28 - University Calculus: Early Transcendentals

Question

Dot multiplication is positive definite Show that dot multiplication of vectors is positive definite; that is, show that \(\mathbf{u} \cdot \mathbf{u} \geq \mathbf{0}\) for every vector \(\mathbf{u}\) and that \(\mathbf{u} \cdot \mathbf{u}=0\) if and only if \(\mathbf{u}=\mathbf{0}\)

Step-by-Step Solution

Answer

The dot product \(\mathbf{u} \cdot \mathbf{u}\) is positive definite because it is non-negative and zero only when \(\mathbf{u} = \mathbf{0}\).

Step 1: Definition of Dot Product

The dot product of a vector \(\mathbf{u} = \begin{bmatrix} u_1 & u_2 & \cdots & u_n \end{bmatrix}^T\) with itself is given by \(\mathbf{u} \cdot \mathbf{u} = u_1^2 + u_2^2 + \cdots + u_n^2\).

Step 2: Show Non-Negativity

Since each term \(u_i^2\) is a square, it is always non-negative. Thus, \(\mathbf{u} \cdot \mathbf{u} = u_1^2 + u_2^2 + \cdots + u_n^2 \geq 0\) for every vector \(\mathbf{u}\).

Step 3: Determine Condition for Equality

The equality \(\mathbf{u} \cdot \mathbf{u} = 0\) holds if and only if each term \(u_i^2 = 0\). This implies that \(u_i = 0\) for all \(i\), meaning \(\mathbf{u} = \mathbf{0}\).

Step 4: Conclusion

Therefore, the dot product \(\mathbf{u} \cdot \mathbf{u}\) is non-negative and equals zero if and only if \(\mathbf{u}\) is the zero vector, confirming that it's positive definite.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector calculus that combines two equal-length sequences of numbers into a single number. Given two vectors \(\mathbf{u} = \begin{bmatrix} u_1 & u_2 & \cdots & u_n \end{bmatrix}^T\) and \(\mathbf{v} = \begin{bmatrix} v_1 & v_2 & \cdots & v_n \end{bmatrix}^T\), the dot product is calculated as:

• \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \cdots + u_nv_n\)

When you perform the dot product of a vector with itself, like \(\mathbf{u} \cdot \mathbf{u}\), each component \(u_i^2\) contributes to the total sum. Since squares of real numbers are always non-negative, the dot product is non-negative. This property plays a crucial role in vector analysis, especially in determining the positivity of expressions.

Positive Definite

A quadratic form (an expression where vectors multiply with each other) is said to be positive definite if the result is always positive, except when the vector itself is zero. Mathematically, for a vector \(\mathbf{u}\), its corresponding quadratic form is positive definite if:

• \(\mathbf{u} \cdot \mathbf{u} \geq 0\)
• \(\mathbf{u} \cdot \mathbf{u} = 0\) if and only if \(\mathbf{u} = \mathbf{0}\)

This simply means that if the vector is not the zero vector, the dot product will yield a positive number. This property is significant in fields like optimization and numerical analysis, as it ensures the stability and convergence of various methods.

Vector Analysis

Vector analysis is the branch of mathematics that deals with vectors and vector operators. It is a tool used predominantly in physics and engineering to describe things like force, velocity, and other vector quantities. Key aspects of vector analysis include operations like the dot product and cross product, which facilitate the understanding of the magnitude and directionality of forces and motions in space.
Understanding vector operations is crucial in fields such as electromagnetism, fluid dynamics, and more. The dot product, in particular, can express projections and compute work done, providing insight into how vectors interact within a multidimensional space.
Vector analysis, by using concepts like positive definiteness, helps ensure that mathematical models accurately reflect physical realities—like ensuring forces behave predictably under transformation.