Lecture 10: Transpose of a Matrix

Introduction

• Lecturer: Nathan Johnston
• Course: Introductory Linear Algebra
• Topic: Transpose of a matrix

Definition of Transpose

• Transpose: Operation interchanging the rows and columns of a matrix.
• Notation: If ( A ) is a matrix, its transpose is denoted by ( A^T ).
• Example:
  • An ( m \times n ) matrix becomes an ( n \times m ) matrix after transposition.
  • Interchange indices: the ( i, j ) entry becomes the ( j, i ) entry.

Visualizing Transpose

• Reflect matrix across its main diagonal.
• Diagonal entries remain in place; off-diagonal entries swap positions.
• Works on non-square matrices as well.

Properties of Transpose

1. Double Transpose:
   • ((A^T)^T = A)
   • Restores the original matrix.
2. Sum of Transposes:
   • ((A + B)^T = A^T + B^T)
   • Entry-wise sum and transpose.
3. Scalar Multiplication:
   • ((kA)^T = kA^T)
   • Scalar multiplication commutes with transpose.
4. Product Transpose:
   • ((AB)^T = B^T A^T)
   • Order of multiplication is reversed.

Proof of Product Transpose Property

• Objective: Show ((AB)^T = B^T A^T).
• Matrix Dimensions:
  • ( A ): ( m \times n )
  • ( B ): ( n \times p )
• Indices:
  • ( i ): row index of ( AB)
  • ( j ): column index of ( AB)
• Process:
  • Show that each entry of ((AB)^T) equals the corresponding entry of (B^T A^T).
  • Use definition of matrix multiplication and transpose.

Practical Examples

• Example Matrices:
  • Transpose matrices ( A ), ( B ), and ( C ).
  • Calculate ( (AB)^T ) and ( B^T A^T ) to verify transposition properties.

Application to Multiple Matrices

• Three Matrices:
  • ((ABC)^T = C^T B^T A^T)
  • General rule: Transpose applied to each matrix, reverse order.

Vectors and Transpose

• Convention: Vectors are considered as column vectors.
• Dot Product:
  • ( v^T w = v \cdot w )
  • Relation between transpose/matrix multiplication and dot product.

Conclusion

• Transpose is a fundamental matrix operation used widely for computations.
• Next class: Discussion on powers of matrices.