Problem 18 - Differential Equations and Linear Algebra

Question

Consider the general \(2 \times 2\) real symmetric matrix \(A=\left[\begin{array}{ll}a & b \\ b & c\end{array}\right] .\) Prove that \(A\) has an eigenvalue of multiplicity two if and only if it is a scalar matrix (that is, a matrix of the form \(r I_{2},\) where \(r\) is a constant).

Step-by-Step Solution

Answer

A general 2x2 real symmetric matrix A has an eigenvalue with multiplicity 2 if and only if it is a scalar matrix. To prove this, we first find the characteristic equation of A: \(\det(A - \lambda I) = (a - \lambda)(c - \lambda) - b^2 = \lambda^2 - (a + c)\lambda + (ac - b^2) = 0\). If A has an eigenvalue with multiplicity 2, then the discriminant of the characteristic equation must be 0. This condition leads to the equation \((a-c)^2 = 4b^2\). To have A as a scalar matrix, we must have a=c and b=0, which satisfies the equation. Conversely, if A is a scalar matrix, then the characteristic equation has a unique solution, giving an eigenvalue with multiplicity 2.

Step 1: Find the characteristic equation of the matrix A

To find the eigenvalues of the given matrix A, we will first find its characteristic equation. It is given by the determinant of (A - λI): \( \det(A - \lambda I) = \det\Big( \begin{bmatrix} a - \lambda & b \\ b & c - \lambda \end{bmatrix} \Big)\). Now, compute the determinant: \(\det(A - \lambda I) = (a - \lambda)(c - \lambda) - b^2\).

Step 2: Solve for the eigenvalues λ

To find the eigenvalues, set the determinant of (A - λI) equal to zero and solve for λ: \((a - \lambda)(c - \lambda) - b^2 = 0\). Expanding, we obtain the quadratic equation: \(\lambda^2 - (a + c)\lambda + (ac - b^2) = 0\).

Step 3: Consider the case when the matrix has an eigenvalue with multiplicity 2

Suppose A has an eigenvalue (λ) with multiplicity 2. This means the quadratic equation has only one distinct solution. The discriminant of the quadratic equation must be zero. This implies the following condition: \((a + c)^2 - 4(ac - b^2) = 0\). Expanding the equation, we get: \(a^2 + 2ac + c^2 = 4ac - 4b^2\) ⟹ \(a^2 + 2ac + c^2 - 4ac + 4b^2 = 0\). Rearranging, we have: \(4b^2 = a^2 - 2ac + c^2\). Now observe that the right-hand side of the equation is equal to the square of the difference between a and c: \((a - c)^2 = a^2 - 2ac + c^2\). Now our equation becomes: \((a-c)^2 = 4b^2\). Since both sides of the equation are non-negative, we must have either (a-c)=2b or (a-c)=-2b. But for the A to be a scalar matrix, it must have the form rI₂, which implies that a=c and b=0. Therefore, for A to have an eigenvalue with multiplicity 2, (a-c)=2b=0. Thus, A must be a scalar matrix.

Step 4: Consider the case when the matrix is a scalar matrix

Now let A be a scalar matrix, that is, a matrix of the form rI₂. In this case, A is given by: \(A = \begin{bmatrix} r & 0 \\ 0 & r \end{bmatrix}\). We will now find the eigenvalues of this matrix. The characteristic equation obtained in step 1 becomes: \((r - \lambda)^2 = 0\). This equation has only one distinct solution, λ=r. Therefore, the eigenvalue λ=r has multiplicity 2. In conclusion, a general 2×2 real symmetric matrix A has an eigenvalue with multiplicity 2 if and only if it is a scalar matrix.

Key Concepts

Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose. This means that the elements of the matrix mirror each other over the main diagonal. For example, consider a matrix \(A\) defined as follows:

• \(A = \begin{bmatrix} a & b \ b & c \end{bmatrix}\)

In this case, the element in the first row and second column is the same as the element in the second row and first column, making the matrix symmetric.
Symmetric matrices have several interesting properties:

• All eigenvalues of a symmetric matrix are real numbers.
• Symmetric matrices are diagonalizable. This means they can be expressed in a form where all non-diagonal elements are zero.

These properties make symmetric matrices significant in various applications, such as optimization problems and physics. The symmetry ensures that complex numbers do not appear as eigenvalues, simplifying calculations.

Characteristic Equation

The characteristic equation of a matrix is a crucial concept when working with eigenvalues. It is derived from the determinant of the matrix subtracted by \(\lambda\) times the identity matrix, represented as \(\det(A - \lambda I) = 0\). For a \(2 \times 2\) symmetric matrix \(A\), the characteristic equation can be detailed as follows:

• Given \(A = \begin{bmatrix} a & b \ b & c \end{bmatrix}\).
• The characteristic determinant is \(\det\begin{bmatrix} a-\lambda & b \ b & c-\lambda \end{bmatrix}\).

Expanding this determinant results in the characteristic equation: \[(a - \lambda)(c - \lambda) - b^2 = 0\].
This is a quadratic equation in \(\lambda\), and solving it gives the eigenvalues of the matrix. When the matrix has a repeated eigenvalue, it indicates special properties about the matrix, such as it potentially being a scalar matrix.

Scalar Matrix

A scalar matrix is a special type of square matrix in which all the elements on the main diagonal are equal, and all off-diagonal elements are zero. In a \(2 \times 2\) format, a scalar matrix \(A\) is expressed as:

• \(A = \begin{bmatrix} r & 0 \ 0 & r \end{bmatrix}\)

where \(r\) is a scalar, or a constant value.

• All scalar matrices are symmetric, but not all symmetric matrices are scalar.
• Scalar matrices have very straightforward eigenvalues. The eigenvalue equation \((r - \lambda)^2 = 0\) shows \(\lambda = r\) with multiplicity equal to the size of the matrix.

In practical applications, scalar matrices conveniently represent scaling operations, where the matrix simply scales any vector it multiplies by the scalar \(r\). This uniform scaling property is why scalar matrices have repeated eigenvalues.

2x2 Matrix

The \(2 \times 2\) matrix is one of the simplest yet powerful structures in linear algebra. Each \(2 \times 2\) matrix contains 4 elements arranged in two rows and two columns. Such matrices can be used to represent a variety of transformations in two-dimensional space, including:

• Rotations
• Reflections
• Scalings
• Shears

For a matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant, \(ad - bc\), is a key feature that tells us whether the matrix is invertible (determinant not equal to zero) and affects the area scaling during transformations. A \(2 \times 2\) symmetric matrix's special structure means its behavior can be neatly summarized by its eigenvectors and eigenvalues, which dictate how it stretches or compresses a space it acts upon. Even though they are miniscule compared to larger matrices, \(2 \times 2\) matrices have distinct and foundational properties that make them critical in learning linear algebra.