Question

If \(A\) is diagonalizable and 1 and -1 are the only eigenvalues, show that \(A^{-1}=A\).

Short answer

If A is diagonalizable with eigenvalues 1 and -1, then \(A^{-1} = A\).

Step-by-step solution

Understanding Diagonalizable Matrices

A matrix \( A \) is diagonalizable if it can be expressed as \( A = PDP^{-1} \), where \( D \) is a diagonal matrix and \( P \) is an invertible matrix. The diagonal elements of \( D \) are the eigenvalues of \( A \). We are given that \( A \) has only 1 and -1 as eigenvalues.

Expressing the Matrix A in Diagonal Form

Since \( A \) is diagonalizable and has eigenvalues 1 and -1, we can express \( D \), the diagonal matrix, as \( D = \text{diag}(1, 1, ..., -1, -1) \) if the matrix \( A \) is \( n \times n \). The number of times 1 and -1 appear on the diagonal corresponds to their algebraic multiplicities.

Finding the Inverse of Diagonal Matrix D

The inverse of a diagonal matrix \( D = \text{diag}(\lambda_1, \lambda_2, ..., \lambda_n) \) is simply \( D^{-1} = \text{diag}(\frac{1}{\lambda_1}, \frac{1}{\lambda_2}, ..., \frac{1}{\lambda_n}) \). Since the eigenvalues are 1 and -1, the inverse \( D^{-1} \) has the same diagonal entries as \( D \), because \( \frac{1}{1} = 1 \) and \( \frac{1}{-1} = -1 \). Thus, \( D^{-1} = D \).

Calculating the Inverse of A

Using \( A = PDP^{-1} \), the inverse of \( A \) is \( A^{-1} = (PDP^{-1})^{-1} = PD^{-1}P^{-1} \). Since we've shown that \( D^{-1} = D \), this gives us \( A^{-1} = PDP^{-1} = A \).

Conclusion

Since the inverse of \( A \), which is \( A^{-1} \), is equal to \( A \) itself, we have shown that \( A^{-1} = A \) holds under the given conditions that 1 and -1 are the only eigenvalues of \( A \).

Key concepts

Eigenvalues

Understanding eigenvalues is essential when studying matrices and how they behave. Eigenvalues are special numbers associated with a matrix, and they give us important information about the matrix's properties.

• An eigenvalue is a scalar value, such that when a specific vector (called the eigenvector) is multiplied by the matrix, it results in the eigenvector being scaled by that scalar.
• If \( \lambda \) is an eigenvalue of a matrix \( A \), then there exists a non-zero vector \( \mathbf{v} \) such that \( A\mathbf{v} = \lambda\mathbf{v} \).
• The eigenvalues of a matrix significantly influence characteristics like its invertibility and trace.
• For a matrix to be diagonalizable, it generally requires enough independent eigenvectors corresponding to its eigenvalues. This is crucial because it allows the matrix to transform into a diagonal form, simplifying many matrix operations.

In our exercise, the given eigenvalues, 1 and -1, make it so that we can express the matrix \( A \) in a diagonal form, revealing patterns and simplifying the process of inversion.

Matrix Inversion

Matrix inversion is a fundamental operation in linear algebra, where we find a matrix that, when multiplied with the original matrix, yields the identity matrix.

• For a square matrix \( A \), the inverse matrix, denoted \( A^{-1} \), satisfies the equation \( AA^{-1} = A^{-1}A = I \), where \( I \) is the identity matrix of the same size as \( A \).
• Not all matrices are invertible. A matrix is invertible only if it has full rank, meaning there are no zero eigenvalues.
• For diagonal matrices, the inversion process is straightforward; simply take the reciprocal of each diagonal element. This works nicely when the eigenvalues are 1 and -1, as these values are their own reciprocals.

In our scenario, the matrix \( A \) is found to be its own inverse since the inverse of the diagonalized form of \( A \), which retains the same eigenvalues, matches the original.

Algebraic Multiplicity

Algebraic multiplicity is a concept used to describe how many times a particular eigenvalue appears in the characteristic polynomial of a matrix.

• The characteristic polynomial of a matrix is obtained by finding the determinant of \( A - \lambda I \), where \( A \) is the matrix in question, \( \lambda \) is a scalar, and \( I \) is the identity matrix.
• The algebraic multiplicity of an eigenvalue \( \lambda \) is essentially the exponent of \( \lambda \) in the factorization of the characteristic polynomial.
• It is possible for an eigenvalue to have both an algebraic multiplicity and a geometric multiplicity, which refers to the number of independent eigenvectors associated with that eigenvalue.
• When a matrix has distinct eigenvalues, it typically means each eigenvalue's algebraic multiplicity is 1.

In our problem, the eigenvalues 1 and -1 have their multiplicity determined by their occurrence in the diagonalized form. This multiplicity helps us confirm that the diagonal matrix \( D \) mirrors \( A \) through its transform, simplifying our calculations.