Question

Prove the following for matrices \(A, B\), and \(C\). a. \((A B) C=A(B C)\). b. \((A B)^{T}=B^{T} A^{T}\) c. \(\operatorname{tr}(A)\) is invariant under similarity transformations. d. If \(A\) and \(B\) are orthogonal, then \(A B\) is orthogonal.

Short answer

a. Matrices associate under multiplication. \n b. Transpose of a matrix product is the product of the transposed matrices, but the order is reversed. \n c. The trace of a matrix is invariant under similarity transformation (cyclic permutation). \n d. The product of two orthogonal matrices is also orthogonal.

Step-by-step solution

Associativity of Matrix Multiplication

Matrix multiplication is associative, which means \((AB)C = A(BC)\). This comes as a result of the definition of matrix multiplication which states that the ith,jth entry of \(AB\) is the sum of the products of elements from the ith row of A and the jth column of B. The same applies to \((AB)C\) and \(A(BC)\). This concludes the proof for part a.

Transpose of a Matrix Multiplication

The transposition of a product of two matrices is equal to the product of their transposes in the reverse order. Mathematically, this is expressed as \((AB)^T = B^T A^T\). The proof of this proposition follows directly from the definition of matrix multiplication and transpose. This rule is used in proving properties of symmetric and orthogonal matrices among other things. This concludes the proof for part b.

Trace Invariance under Similarity Transformations

The trace of a matrix is invariant under similarity transformations, meaning for a matrix \(A\) and an invertible matrix \(P\), \(\operatorname{tr}(A) = \operatorname{tr}(P^{-1}AP)\). This is because trace has the property of being invariant under cyclic permutations, meaning the trace of the product of matrices is the same no matter how they are grouped. This concludes the proof for part c.

Orthogonality of Matrices

The final part of this problem involves ordinal matrices, which have the property \(AA^T = A^T A = I\). If \(A\) and \(B\) are both orthogonal, then their product \(AB\) is also orthogonal. We prove this by showing that \((AB)(AB)^T = ABB^TA^T = AA^T = I\), and \((AB)^T(AB) = B^TA^T AB=B^TBA=I\). This finalizes the proof for d.

Key concepts

Associativity in Matrix Multiplication

Understanding the concept of associativity in matrix multiplication is pivotal for students grappling with linear algebra. Just as addition and multiplication of numbers are associative, so too is matrix multiplication. What this means in practice is that no matter how we group matrices during multiplication, the result remains unchanged.

Let's take matrices A, B, and C as an example. When we calculate \( (AB)C \) and \( A(BC) \) separately, both of these products will yield the same result. This characteristic is extremely helpful because it allows us to simplify complex matrix equations and compute them in any order we prefer without altering the outcome. This comes directly from the way matrices are multiplied, where elements are combined in a specific sum-product manner to form new elements. It's also why this property is taken for granted; it underpins more complicated operations in linear algebra and ensures their correctness.

Transpose of a Matrix

Flipping a matrix over its diagonal to switch its rows and columns results in what we call the transpose of a matrix. This operation is denoted as \( A^T \) for the transpose of matrix A. When we multiply two matrices together and then take the transpose of the result, a fascinating switch takes place: \( (AB)^T = B^T A^T \).

What's surprising here is the reversal of order—what was once \( AB \) becomes \( B^T A^T \) post transposition. This property is incredibly useful in the realm of mathematical proofs and operations, particularly for symmetric and orthogonal matrices. Understanding the behavior of matrix transposition is crucial for tackling problems involving matrix equations and for recognizing patterns within matrices relating to their symmetry and other attributes.

Trace Invariance under Similarity Transformations

The trace of a matrix, the sum of the diagonal elements, plays a special role in matrix theory due to its inherent properties. One of these is the invariance under similarity transformations, a term that might perplex students at first. To unravel this, consider a matrix A and an invertible matrix P; a similarity transformation involves the operation \( P^{-1}AP \). The beauty lies in the fact that the trace of A remains the same even after such a transformation, formally stated as \( \operatorname{tr}(A) = \operatorname{tr}(P^{-1}AP) \).

This invariant property of trace is significant in the study of eigenvalues and other areas where similarity transformations are common. The reason behind this invariance is that the trace is unaffected by the cyclic permutations of matrix products. Retaining this property through various transformations makes the trace a valuable tool in the analysis of matrix characteristics and behaviors.

Orthogonality of Matrices

Delving into the realm of orthogonality in matrices introduces students to one of the more sublime concepts in linear algebra. A matrix is orthogonal if, when multiplied by its transpose, the result is the identity matrix \( I \). Technically put, for matrix A, if \( AA^T = A^T A = I \), A is orthogonal. An immediate corollary of this is that if two matrices A and B are orthogonal, their product \( AB \) is also orthogonal.

Orthogonal matrices are a keystone in many areas of mathematics and engineering because of their stability and the preservation of vector norms during transformations. Whether one is rotating points in space, stability and simplicity result when orthogonal transformations are at work. Recognizing the instances where orthogonality can be utilized to simplify problems or deduce properties of matrix operations is an essential skill for students to develop.