Question

If \(A\) is diagonalizable, show that each of the following is also diagonalizable. a. \(A^{n}, n \geq 1\) b. \(k A, k\) any scalar. c. \(p(A), p(x)\) any polynomial (Theorem 3.3.1) d. \(U^{-1} A U\) for any invertible matrix \(U\). e. \(k I+A\) for any scalar \(k\).

Short answer

All matrices given are diagonalizable if \(A\) is diagonalizable.

Step-by-step solution

Understanding Diagonalizability

A matrix \( A \) is diagonalizable if there exists an invertible matrix \( P \) such that \( P^{-1}AP = D \), where \( D \) is a diagonal matrix. This property implies that the matrix can be represented in a form where its eigenvalues are on the diagonal, making operations easier.

Prove that \(A^{n}\) is Diagonalizable

Since \( A \) is diagonalizable, let \( A = PDP^{-1} \) where \( D \) is diagonal. Then \( A^n = (PDP^{-1})^n = PD^nP^{-1} \). Since \( D^n \) is also a diagonal matrix, \( A^n \) remains diagonalizable.

Prove that \(kA\) is Diagonalizable

Given \( A = PDP^{-1} \), multiplying by a scalar \( k \) gives \( kA = kPDP^{-1} = P(kD)P^{-1} \). Since \( kD \) is a diagonal matrix, \( kA \) is diagonalizable.

Prove that \(p(A)\) is Diagonalizable

If \( A = PDP^{-1} \), then \( p(A) = Pp(D)P^{-1} \) by substituting \( D \). Since \( p(D) \) is still diagonal (based on evaluating a diagonal matrix with polynomial \( p \)), \( p(A) \) is diagonalizable.

Prove that \(U^{-1} A U\) is Diagonalizable

For any invertible matrix \( U \), if \( A = PDP^{-1} \), then \( U^{-1}AU = U^{-1}(PDP^{-1})U = (U^{-1}P)D(P^{-1}U) \). Since this fits the form \( M^{-1}DM \), it is diagonalizable.

Prove that \(kI + A\) is Diagonalizable

As \( A = PDP^{-1} \), then \( kI + A = P(kI + D)P^{-1} \). Since \( kI + D \) is diagonal (a scalar identity matrix added to a diagonal matrix), \( kI + A \) is diagonalizable.

Key concepts

Matrix Diagonalization

Matrix diagonalization is a powerful tool in linear algebra. It involves transforming a given matrix into a diagonal one. This process requires finding an invertible matrix \( P \) such that \( P^{-1}AP = D \). Here, \( D \) denotes a diagonal matrix. The beauty of diagonalization lies in its ability to simplify matrix operations. Diagonal matrices are easier to work with since their non-zero entries are confined to the diagonal. Thus, when a matrix is diagonalizable, complex computations like matrix powers become significantly easier to handle.

Eigenvalues

The concept of eigenvalues is central to understanding matrix diagonalization. When dealing with diagonalizable matrices, the eigenvalues are positioned along the diagonal in the matrix \( D \). An eigenvalue is a special kind of scalar associated with a matrix. It ensures that multiplying a matrix by a vector results in the vector's scaling, rather than changing its direction.

• To find eigenvalues, you typically solve the characteristic equation \( \det(A - \lambda I) = 0 \), where \( \lambda \) stands for the eigenvalues.
• Eigenvalues offer insights into the properties of a matrix. For instance, they tell us about the stability and potential transformation a system can undergo.
• For diagonalizable matrices, the eigenvalues strictly correspond to the diagonal entries in the matrix \( D \).

Polynomial Matrices

Polynomial matrices involve applying polynomial functions to matrices. If we have a polynomial \( p(x) \), and a matrix \( A \), we can compute \( p(A) \) for functions like \( x^2 + 3x + 2 \). This means substituting the operations directly applied to the scalar \( x \) onto the matrix \( A \).

For a diagonalizable matrix, when we calculate \( p(A) \), it turns into \( Pp(D)P^{-1} \). Here, \( p(D) \) involves simply evaluating the polynomial at each of the eigenvalues (the diagonal entries of \( D \)). The polynomial matrix \( p(A) \) remains diagonalizable as long as the original matrix \( A \) is diagonalizable. Thus, polynomial operations maintain ease and efficiency because the polynomial transformation translates directly to the diagonal entries.

Invertible Matrices

An invertible matrix, or non-singular matrix, is a square matrix that has an inverse. The key property here is that if a matrix \( A \) is invertible, there exists another matrix \( A^{-1} \) such that \( AA^{-1} = I \), where \( I \) is the identity matrix. This property plays a crucial role in diagonalization.

When discussing diagonalization, attempting to transform a matrix \( A \) into the form \( U^{-1}AU \) for any invertible matrix \( U \) is possible without losing the diagonalizable property. This operation essentially changes the basis in which \( A \) expresses itself, while retaining its inherent structure since the transformation \( (U^{-1}P)D(P^{-1}U) \) maintains the diagonal form \( (M^{-1}DM) \). Thus, it all ties back into ensuring computational simplicity for higher-dimensional operations while preserving core characteristics of the matrix.