Question

Show that the only matrix similar to a scalar matrix \(A=r I, r\) in \(\mathbb{R},\) is \(A\) itself.

Short answer

The only matrix similar to a scalar matrix \( A = rI \) is \( A \) itself.

Step-by-step solution

Define Scalar Matrix and Similar Matrices

A scalar matrix is a special type of diagonal matrix where all the diagonal elements are equal, denoted as \( A = rI \), where \( r \) is a scalar and \( I \) is the identity matrix. Matrix \( B \) is similar to \( A \) if there exists an invertible matrix \( P \) such that \( B = P^{-1}AP \). Our aim is to show that \( B = A = rI \).

Simplify P^{-1}AP for Scalar Matrix

Since \( A = rI \), we have \( P^{-1}AP = P^{-1}(rI)P = rP^{-1}IP = rP^{-1}P \). The identity matrix \( I \) does not change the matrix \( P \), so \( r P^{-1}IP = rI \). Thus, \( P^{-1}AP = rI \).

Establish that Similar Matrices Have the Same Eigenvalues

Eigenvalues are invariant under similarity transformations. The eigenvalues of \( A \) (which is \( rI \)) are \( r \) repeated for the dimension of \( I \). Thus, any matrix similar to \( A \) must also have only \( r \) as its eigenvalue, with the same multiplicity.

Prove Matrix Equality

Since \( P^{-1}AP = rI = A \) from calculation and shares identical eigenvalues, \( B = rI \). Therefore, the only matrix similar to \( A = rI \) is \( A \) itself, i.e., \( B \) cannot differ from \( A \).

Conclude with Matrix Equality Proof

For any invertible matrix \( P \), \( P^{-1}AP = A \) proves that \( A \) is similar only to itself because no other matrices satisfy both the algebraic and eigenvalue conditions of \( A \). Therefore, the proof shows that the only matrix similar to a scalar matrix is the matrix itself.

Key concepts

Scalar Matrix

A scalar matrix is a particular type of diagonal matrix where all the elements on the main diagonal are the same constant value, denoted by a scalar.
It is written in the form of \( A = rI \), where \( r \) is a real number termed as the scalar, and \( I \) is the identity matrix.

• Each diagonal element is equal to \( r \).
• All off-diagonal elements are zero.
• An example of a scalar matrix for a 3x3 matrix would be \[ r \cdot\begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} r & 0 & 0 \ 0 & r & 0 \ 0 & 0 & r \end{pmatrix} \]

In essence, a scalar matrix is just a multiple of the identity matrix, scaling it by multiplying each diagonal by \( r \). This uniformity simplifies many matrix operations involving scalar matrices.

Eigenvalues

Eigenvalues are specific values which show how a linear transformation described by a matrix can scale vectors without changing their direction.
For scalar matrices, determining eigenvalues becomes straightforward. If you have a scalar matrix \( A = rI \), its eigenvalues are simply \( r \) repeated \( n \) times, where \( n \) is the size/dimension of the matrix.

• These eigenvalues are computed by solving the characteristic equation \( \det(A - \lambda I) = 0 \).
• For a scalar matrix \( A \), it transforms into \( \det(rI - \lambda I) = (r - \lambda)^n = 0 \).
• So, \( \lambda = r \) is the eigenvalue, repeated \( n \) times.

Eigenvalues being invariant under similarity transformations ensures that any matrix similar to our scalar matrix \( A = rI \) must have these same eigenvalues. This property plays a crucial role in proving that scalar matrices are only similar to themselves.

Identity Matrix

An identity matrix, typically denoted as \( I \) or \( I_n \) for an \( n imes n \) matrix, is a square matrix with ones on the main diagonal and zeros elsewhere.
This matrix functions as the equivalent of "1" in matrix algebra.

• Any matrix \( A \) multiplied by the identity matrix results in \( A \) itself. ( \( AI = IA = A \))
• For any real number \( r \) and an identity matrix \( I \), \( rI \) turns the identity matrix into a scalar matrix by multiplying each diagonal element by \( r \).

The role of the identity matrix in our problem is pivotal because it preserves the matrix form when matrices multiply, specifically helping to simplify expressions like \( P^{-1}AP \) to \( rI \). This preservation demonstrates that the transformation does not alter the scalar matrix \( A \), affirming its uniqueness in similarity.