Question

If \(A\) is an \(n \times n\) matrix, show that \(A\) is diagonalizable if and only if \(A^{T}\) is diagonalizable.

Short answer

\(A\) is diagonalizable if and only if \(A^T\) is diagonalizable.

Step-by-step solution

Understand the Definitions

A matrix is diagonalizable if there exists an invertible matrix \(P\) such that \(P^{-1}AP\) is a diagonal matrix. Similarly, \(A^T\) being diagonalizable means there exists an invertible matrix \(Q\) such that \(Q^{-1}A^TQ\) is a diagonal matrix.

Prove The Forward Implication

Assume \(A\) is diagonalizable. This means there exists \(P\) such that \(P^{-1}AP = D\), where \(D\) is a diagonal matrix. Since the transpose of a diagonal matrix is itself, we have \((P^{-1}AP)^T = D^T = D\). Therefore, \(P^T A^T (P^{-1})^T = D\). Let \(Q = (P^{-1})^T\), then \((Q^T)^{-1} A^T Q^T = D\). Thus, \(A^T\) is diagonalizable as well.

Prove The Reverse Implication

Assume \(A^T\) is diagonalizable. Therefore, there exists an invertible matrix \(Q\) such that \(Q^{-1}A^TQ = D\), where \(D\) is a diagonal matrix. Taking the transpose gives \((Q^{-1}A^TQ)^T = D^T = D\). This implies \(Q^T A (Q^{-1})^T = D\). Let \(P = Q^T\), then \(P^{-1}AP = D\) showing \(A\) is also diagonalizable.

Summarize the Conclusion

We've shown that if \(A\) is diagonalizable, then \(A^T\) is diagonalizable and vice versa. Thus, \(A\) is diagonalizable if and only if \(A^T\) is diagonalizable.

Key concepts

Invertible Matrix

An invertible matrix, also known as a non-singular or non-degenerate matrix, is a square matrix that has an inverse. This means there exists another matrix, often denoted as \(P^{-1}\) for a given matrix \(P\), such that when they are multiplied together, they yield the identity matrix. The identity matrix acts like the number 1 in matrix multiplication, leaving other matrices unchanged.
What makes an invertible matrix special? Here are the key attributes:

• For a matrix \(P\), if \(P \cdot P^{-1} = P^{-1} \cdot P = I\) (where \(I\) is the identity matrix), then \(P\) is invertible.
• To determine if a matrix is invertible, its determinant (a scalar value) must not be equal to zero.
• If a matrix isn't invertible, it doesn't have an inverse and is called singular.

Understanding invertible matrices is crucial because they facilitate various operations in linear algebra, such as solving systems of linear equations and performing matrix diagonalization.

Transpose of a Matrix

The transpose of a matrix is a simple yet important operation in linear algebra. For a given matrix \(A\), the transpose, denoted as \(A^T\), is formed by swapping its rows and columns. If matrix \(A\) is represented by a series of rows, \(A^T\) changes these rows into columns and vice versa.
Let's look at some properties of transposes:

• The transpose of a transpose brings us back to the original matrix: \((A^T)^T = A\).
• Transposing a diagonal matrix doesn't change it, which is particularly relevant to diagonalization.
• The transpose of a product of two matrices \(A\) and \(B\) is the product of their transposes in reverse order: \((AB)^T = B^T A^T\).

Understanding the transpose is key to many matrix manipulations, especially those involving symmetry and orthogonality.

Diagonal Matrix

A diagonal matrix is a type of square matrix where all elements outside the main diagonal are zero. The principal diagonal from the top-left to the bottom-right can have non-zero elements, and those are the only non-zero entries allowed in a diagonal matrix. Diagonal matrices are simple structures that play a big role in matrix diagonalization.
Here are some useful properties of diagonal matrices:

• Addition and multiplication of diagonal matrices is straightforward because you can treat each diagonal entry separately.
• The power of a diagonal matrix is easy to compute: each diagonal element is raised to the power, forming the corresponding element of the resulting matrix.
• An important aspect of a diagonal matrix is that it's both upper and lower triangular, meaning calculations involving determinants and inverses are simplified.

Diagonal matrices are essential in simplifying complex operations with matrices, making them vital tools for theoretical mathematics and practical applications like computer graphics and scientific computations.