29.4. Properties of Eigenvalues

• The sum of eigenvalues (counted with multiplicity) is equal to the trace of the matrix: $\sum \lambda = \text{tr}(A)$
• The product of eigenvalues is equal to the determinant: $\prod \lambda = \text{det}(A)$

This chunk explains two important properties of eigenvalues related to the trace and determinant of a matrix.

1. Trace: The trace of a matrix, denoted as tr(A), is the sum of the elements on its main diagonal. When considering a square matrix, the sum of its eigenvalues (considering their multiplicities) equals the trace. This means that if you add up all the eigenvalues, it will give you the same result as the trace.

2. Determinant: The determinant of a matrix is a scalar value that represents certain properties of the matrix, such as whether it is invertible. The product of all eigenvalues gives you the determinant of the matrix. In other words, if you multiply all the eigenvalues together, it will equal the determinant of the original matrix.

Imagine a classroom where each student's score contributes to the total score of the class. The trace is comparable to this total score— it reflects how well the class is doing overall. On the other hand, the determinant is like the overall health of the class: if the score is very low, it could indicate issues like students failing to understand the material. Thus, the scores (eigenvalues) give insights into the performance (trace) and solvability (determinant) of the class.

• For an upper or lower triangular matrix, the eigenvalues are simply the diagonal entries.

This chunk focuses on a specific type of matrix: triangular matrices. There are two types, upper triangular and lower triangular matrices.

1. Upper Triangular Matrix: This is a matrix where all the entries below the main diagonal are zero. The eigenvalues of this type of matrix are the values found on the diagonal (the entries in the top-left to bottom-right path of the matrix).

2. Lower Triangular Matrix: Conversely, in a lower triangular matrix, all entries above the main diagonal are zero, and similarly, its eigenvalues are the diagonal entries.

Think of a staircase for a triangular matrix: each step represents a diagonal entry that can be directly touched or accessed. The steps below (for an upper triangular matrix) or above (for a lower triangular matrix) are unavailable, indicating that the eigenvalues are just those visible on that diagonal, similar to how you can only see and interact with the steps directly in front of you.

• Have real eigenvalues.
• Are orthogonally diagonalizable (important in principal stress/strain analysis).

This chunk discusses two critical properties of real symmetric matrices, which are matrices that are equal to their transpose.

1. Real Eigenvalues: One significant property of real symmetric matrices is that their eigenvalues are always real numbers (as opposed to complex numbers). This is important for practical applications in engineering and physics, where negative or complex values can be nonsensical in certain contexts.

2. Orthogonal Diagonalization: A real symmetric matrix can be diagonalized using an orthogonal matrix. This means that it can be transformed into a diagonal matrix using an orthogonal transformation (meaning the transpose of the matrix is equal to its inverse). This property is particularly valuable in fields like civil engineering, especially during stress and strain analysis.

Consider a well-built arch bridge. The arch's shape represents a real symmetric matrix, providing real 'support' (which correlates to real eigenvalues). When engineers analyze the forces within the bridge, they use the principle of orthogonal diagonalization to simplify their calculations, much like how one would analyze the forces on a simple structure instead of an overly complex one by breaking it down into wooden blocks representing simpler shapes.

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