A vector has one scalar invariant-its magnitude or length-when its components are transformed by a rotation of axes. Show that a symmetric dyadic \(S\) is characterized by three scalar invariants. \((a)\) its trace, \(I_{1}=S_{11}+S_{22}+S_{33} ;(b)\) the sum of its diagonal minors, $$ I_{2}=\left|\begin{array}{ll} S_{11} & S_{12} \\ S_{21} & S_{12} \end{array}\right|+\left|\begin{array}{ll} S_{22} & S_{18} \\ S_{32} & S_{33} \end{array}\right|+\left|\begin{array}{ll} S_{33} & S_{21} \\ S_{13} & S_{11} \end{array}\right| $$ (c) its delerminanl, \(I_{3}=\left|S_{i j}\right|\).

The trace of a matrix is very much akin to the heart rate for humans; it's a vital measurement that gives insight into the matrix's condition. For a square matrix—the only type where trace is defined—it is simply the sum of the diagonal elements. In the context of our symmetric dyadic matrix, the trace, denoted as \( I_1 \), is calculated by summing the elements \( S_{11} \), \( S_{22} \), and \( S_{33} \).

This invariant is significant because it remains constant under any rotation of axes. Just as people's heart rate doesn't change whether they're standing or lying down, the trace of a matrix stays the same regardless of how we rotate our coordinate system. It is an intrinsic property that provides meaningful information about the matrix's characteristics, including the average eigenvalue and the amount of 'stretch' in the transformation it represents.

Imagine you're looking at a woven fabric pattern, focusing on how the threads interlace to form the design—the individual patterns reflect the entire fabric's structure. Similarly, diagonal minors offer a glimpse into the larger picture of a matrix's structure. Diagonal minors of a matrix are the determinants of the smaller, 2x2 matrices formed when you remove corresponding row and column of each diagonal element one at a time.

In our symmetric dyadic matrix \( S \), we calculate three diagonal minors, each omitting one of \( S \)'s principal diagonal elements, and then sum them up to get \( I_2 \). These minors give us valuable information about the 'subpatterns' or substructures within the symmetrical matrix. They too remain unchanged under a rotation, much like how the essential details of a fabric pattern would persist no matter how you turn the cloth.

The determinant is like a multidimensional volume measuring tool for matrices and serves as a litmus test for various properties, like whether a set of vectors is linearly dependent or the matrix is invertible. For our symmetric dyadic matrix \( S \), the determinant, which we call \( I_3 \) in this specific exercise, represents a scalar value that encapsulates the entire matrix's essence.

To compute it, we expand the matrix along any row or column, in a method known as cofactor expansion, to reduce the matrix to smaller, more manageable pieces until we are left with a single number. This operation simplifies complex, multidimensional relationships into a quantifiable attribute that, just like a 3D object's volume, stays the same regardless of how it's rotated or viewed, thereby qualifying as a scalar invariant of the matrix. Understanding and calculating the determinant is crucial for many areas of mathematics and physics, as it appears in solving linear systems, changing variable in integrals, and more.