Question

Assume that \(M\) is symmetric, \(C\) is orthogonal, and \(D=C^{-1} M C\) is diagonal. Show that the sum of the squares of the elements of \(M\) cquals the sum of the squares of its eigenvalues. Hint: Consider \(\operatorname{Tr}\left(D^{2}\right)\)

Short answer

\(\sum m_{ij}^2 = \sum \lambda_i^2 \)

Step-by-step solution

Understand the problem

Given a symmetric matrix \(M\) and an orthogonal matrix \(C\), we know that \(D = C^{-1} M C\) is a diagonal matrix. We need to show that the sum of the squares of the elements of \(M\) equals the sum of the squares of its eigenvalues.

Recall properties of matrices

Recall that for an orthogonal matrix \(C\), \(C^{-1} = C^T\) and for a symmetric matrix \(M\), all eigenvalues are real. Since \(D\) is diagonal, it consists of the eigenvalues of \(M\).

Express \(M\) using eigenvalues

Since \(M\) is symmetric, it can be diagonalized by an orthogonal matrix \(C\): \(M = C D C^T\). Here, \(D\) is the diagonal matrix whose diagonal elements are the eigenvalues of \(M\).

Trace relationship

The trace of a matrix is the sum of its diagonal elements. The trace of a matrix is invariant under similarity transformations. Therefore, \(\operatorname{Tr}(M^2) = \operatorname{Tr}((C D C^T)^2)\).

Simplify using orthogonality

Using the orthogonality of \(C\), \((C D C^T)^2 = C D^2 C^T\). Hence, \(\operatorname{Tr}(M^2) = \operatorname{Tr}(C D^2 C^T)\). Since trace is invariant under such transformations, \(\operatorname{Tr}(C D^2 C^T) = \operatorname{Tr}(D^2)\).

Sum of squares

The sum of the squares of the elements of \(M\) is \(\operatorname{Tr}(M^2)\). On the other hand, the sum of the squares of the eigenvalues of \(M\) is \(\operatorname{Tr}(D^2)\). Since \(\operatorname{Tr}(M^2) = \operatorname{Tr}(D^2)\), the sum of the squares of the elements of \(M\) equals the sum of the squares of its eigenvalues.

Key concepts

symmetric matrix

A symmetric matrix is a square matrix that is equal to its transpose. This simply means that the matrix remains unchanged when mirrored along its main diagonal.
For example, if you have a 2x2 symmetric matrix:
a =

orthogonal matrix

A matrix is orthogonal if its transpose is also its inverse. This property is very useful for computations because it preserves lengths and angles. In simpler terms, for an orthogonal matrix C: C^T = C^{-1} where C^T is the transpose of C and C^{-1} is the inverse of C.

• This means multiplying an orthogonal matrix by its transpose yields the identity matrix, I (I is a matrix with 1s on the diagonal and 0s elsewhere).
• Orthogonal matrices are very important in transformations and preserving the geometry of data.

eigenvalues

Eigenvalues are the special numbers associated with a square matrix that help in understanding many of its properties. If you have a matrix M, an eigenvalue \(\lambda\) and corresponding eigenvector v satisfy the equation: M*V = \lambdaV

• This means that when we apply matrix M to vector v, the result is a scalar multiple of v (no change in direction).
• The eigenvalues of a symmetric matrix are always real numbers.

trace of a matrix

The trace of a matrix is the sum of its diagonal elements. The trace is denoted as Tr(M) for a matrix M. An important property of trace is that it is invariant under similarity transformations. In other words, for matrices A and B, if A and B are similar (i.e., there exists a matrix C such that A = C^{-1}BC), then Tr(A) = Tr(B).
This property is used in the exercise to show that the sum of the squares of the elements of M is equal to the sum of the squares of its eigenvalues. Specifically:

• Tr(M) = Tr(C^{-1}MC)
• Thus, Tr(M^2) = Tr((C^{-1}MC)^2) = Tr(C^{-1}MCM'C^{-1}) = Tr(D^2).

In conclusion, the trace helps tie together the characteristics of a matrix and its eigenvalues.