Question

Assume that \(A\) and \(B\) are orthogonally similar (Exercise 8.2.12). a. If \(A\) and \(B\) are invertible, show that \(A^{-1}\) and \(B^{-1}\) are orthogonally similar. b. Show that \(A^{2}\) and \(B^{2}\) are orthogonally similar. c. Show that, if \(A\) is symmetric, so is \(B\).

Short answer

If matrices A and B are orthogonally similar: a. Their inverses are also orthogonally similar. b. Their squares are orthogonally similar. c. Symmetry of A implies symmetry of B.

Step-by-step solution

Understanding Orthogonal Similarity

Two matrices \( A \) and \( B \) are orthogonally similar if there exists an orthogonal matrix \( Q \) such that \( B = Q^T A Q \). An orthogonal matrix \( Q \) satisfies \( Q^T Q = I \), where \( I \) is the identity matrix.

Showing Inverses are Orthogonally Similar

Given that \( A \) and \( B \) are orthogonally similar, \( B = Q^T A Q \). If \( A \) and \( B \) are invertible, then \( A^{-1} = Q B^{-1} Q^T \) since multiplying both sides by the inverse relation we have \( A^{-1} = (Q^T A Q)^{-1} = Q B^{-1} Q^T \). Thus, \( A^{-1} \) and \( B^{-1} \) are orthogonally similar.

Showing Squares are Orthogonally Similar

With \( B = Q^T A Q \), we square both sides to get \( B^2 = (Q^T A Q)(Q^T A Q) = Q^T A^2 Q \). Thus, \( A^2 \) and \( B^2 \) are orthogonally similar because \( B^2 = Q^T A^2 Q \).

Showing Symmetric Property is Preserved

If \( A \) is symmetric, \( A = A^T \). Knowing \( B = Q^T A Q \), then \( B^T = (Q^T A Q)^T = Q^T A^T Q = Q^T A Q = B \). Therefore, if \( A \) is symmetric, then \( B \) is also symmetric, preserving symmetry in orthogonal similarity.

Key concepts

Symmetric Matrices

Symmetric matrices are an interesting class of matrices where the elements are mirrored along the diagonal. This means for any matrix \( A \) that is symmetric, we have \( A = A^T \). This property indicates that the matrix looks the same whether you read its rows or its columns.
Looking back at the exercise, if matrix \( A \) is symmetric and \( A \) is orthogonally similar to \( B \), then \( B \) also inherits this property. This is because when you transpose an orthogonally similar matrix transformation: \( B = Q^T A Q \), when it involves a symmetric matrix, it naturally remains symmetric. In other words:

• \( A = A^T \) implies \( B = B^T \)
• The orthogonal matrix \( Q \) preserves symmetry

Therefore, symmetry in matrices is robust and maintained even under orthogonal transformation.

Matrix Inverse

The inverse of a matrix, denoted as \( A^{-1} \), is a matrix that when multiplied by the original matrix \( A \), results in the identity matrix \( I \). In mathematical terms, \( AA^{-1} = I \) and \( A^{-1}A = I \).
This property is particularly useful in solving systems of linear equations among other applications.
In the context of orthogonal similarity, if two matrices \( A \) and \( B \) are orthogonally similar and both invertible, their inverses \( A^{-1} \) and \( B^{-1} \) also share this similarity. This is because the inverse relationship maintains the structure of similarity through:

• \( A^{-1} = Q B^{-1} Q^T \)
• The orthogonality of matrix \( Q \) ensures \( Q^T Q = I \), preserving the structural integrity

This ensures that not only are the original matrices similar, but their operations and transformations are as well.

Matrix Multiplication

Matrix multiplication involves the process of dotting rows from one matrix with columns of another, resulting in a new matrix. This process is foundational in many areas such as computer science, physics, and engineering.
In mathematical terms, if you have two matrices, \( A \) (\( m \times n \)) and \( B \) (\( n \times p \)), their multiplication yields a matrix \( C \) of dimensions \( m \times p \).
When applied to orthogonally similar matrices, the operation confirms their retained similarity—for example, when multiplying to find squares:

• \( B^2 = (Q^T A Q)(Q^T A Q) = Q^T A^2 Q \)
• This showcases that matrix multiplication adheres to the properties of orthogonality

Thus, squaring matrices in this way demonstrates they are still linked by the original orthogonal transformation.

Orthogonal Matrices

Orthogonal matrices play a crucial role in linear algebra, especially concerning transformations that preserve angles and lengths. A matrix \( Q \) is orthogonal if its transpose is equal to its inverse, i.e., \( Q^T = Q^{-1} \). Furthermore, this means \( Q^TQ = I \), where \( I \) is the identity matrix.
Orthogonality is significant when discussing similarity of matrices. Two matrices \( A \) and \( B \) are orthogonally similar if \( B = Q^T A Q \) with an orthogonal matrix \( Q \). This similarity keeps the matrix properties intact:

• Preservation of matrix determinants
• No change in eigenvalues

The orthogonal nature of \( Q \) maintains a stable transformation environment, ensuring that essential features of the matrices are preserved across transformations.