Cauchy-Schwarz Inequality in Excel

The Cauchy-Schwarz Inequality is a mathematical statement that relates the lengths and angles of two vectors. A vector is a quantity that has both magnitude and direction, such as displacement, velocity, or force. The length of a vector is the square root of the sum of the squares of its components, and the angle between two vectors is the inverse cosine of the ratio of their dot product and the product of their lengths.

The Cauchy-Schwarz Inequality says that the absolute value of the dot product of two vectors is always less than or equal to the product of their lengths. In other words, the cosine of the angle between two vectors is always between -1 and 1. This makes sense, because the angle between two vectors can never be more than 180 degrees or less than 0 degrees.

Basic Theory:

The Cauchy-Schwarz Inequality is a mathematical concept that establishes a relationship between the inner product of two vectors and their magnitudes. For two vectors, $\mathbf{a} = (a_1, a_2, \ldots, a_n)$ and $\mathbf{b} = (b_1, b_2, \ldots, b_n)$ , the Cauchy-Schwarz Inequality is given by:

$\left( \sum_{i=1}^{n} a_i b_i \right)^2 \leq \left( \sum_{i=1}^{n} a_i^2 \right) \left( \sum_{i=1}^{n} b_i^2 \right)$

This inequality provides a bound on the dot product of two vectors in terms of their magnitudes. Equality holds if the vectors are proportional.

Procedures in Excel:

Data Input: Begin by entering your vectors in two separate columns. For example, use columns A and B for vectors $\mathbf{a}$ and $\mathbf{b}$ , respectively.
Calculate Dot Product: In an empty cell, use the formula =SUMPRODUCT(A2:A6, B2:B6) to compute the dot product of the two vectors.
Calculate Magnitudes: In two separate cells, calculate the magnitudes of each vector using the formulas =SQRT(SUMSQ(A2:A6)) and =SQRT(SUMSQ(B2:B6)).
Apply Cauchy-Schwarz Inequality: In another cell, square the dot product calculated in step 2. Then, compare it with the product of the magnitudes squared. The formula would be similar to =C2^2 <= D2*E2.

Scenario:

Let's consider a scenario where $\mathbf{a} = (2, 3, 4)$ and $\mathbf{b} = (1, -1, 2)$ .

Vector $\mathbf{a}$	Vector $\mathbf{b}$
2	1
3	-1
4	2

Calculation:

Dot Product: =SUMPRODUCT(A2:A4, B2:B4) results in 12.
Magnitude of $\mathbf{a}$ : =SQRT(SUMSQ(A2:A4)) results in approximately 5.39.
Magnitude of $\mathbf{b}$ : =SQRT(SUMSQ(B2:B4)) results in approximately 2.24.
Applying the Cauchy-Schwarz Inequality: $12^2 \leq 5.39 \times 2.24 \times 2.24$ is TRUE.

Result:

The Cauchy-Schwarz Inequality holds true for the given vectors.

Alternative Approach:

Another approach to applying the Cauchy-Schwarz Inequality in Excel involves using array formulas. This allows for a more concise representation of the calculations, especially for larger datasets.