Transpose of a Matrix

A matrix M morphs into its transpose M^T through a simple yet powerful transformation: it exchanges its rows with columns.

To elucidate, an element nestled in the i-th row and j-th column of the original matrix magically migrates to the j-th row and i-th column of the transposed matrix.

Diving into a Practical Example of Transposed Matrix

Consider for a moment the matrix A:

$$ A= \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} $$

This is a 2x3 matrix, with a composition of two rows and three columns.

To shape its transposed counterpart, all we do is elegantly interchange the rows and columns.

 

$$ A^T= \begin{pmatrix} 1 & 4 & \\ 2 & 5 \\ 3 & 6 \end{pmatrix} $$

This metamorphosis results in the transposed matrix A^T, where:

• The first row of matrix A transforms into the first column of the transposed matrix A^T.
• The second row of matrix A evolves into the second column of the transposed matrix A^T.

Speaking in general terms, if A is an m x n matrix, then its transpose A^T is an n x m matrix, where each element (i, j) in AT directly corresponds to the element (j, i) in A.

Properties of Transpose Matrices

Transpose matrices reveal a number of intriguing properties.

Here are the key ones for consideration:

1. Double transposition returns you to your original matrix. In essence, performing the transpose operation twice lands you back at your point of departure. $$ (A^T)^T = A $$

2. The transpose of a sum of two matrices equates to the sum of their respective transposes. $$ (A+B)^T = A^T+B^T $$

3. After multiplying two matrices and transposing, you'll find an identical result as if you'd initially transposed the matrices and then performed multiplication in a reversed order. $$ (AB)^T = B^TA^T $$

4. This principle extends elegantly to numerous matrices: the transpose of the product of several matrices parallels the product of their transposes, only taken in reverse order. $$ (ABC)^T = C^TB^TA^T $$

5. If you multiply a matrix by a scalar k, then transpose, you'll discover the same outcome as if you'd initially transposed the matrix and subsequently multiplied it by the scalar. $$ (kA)^T = kA^T $$

6. The determinant of a square matrix and its transpose are beautifully equivalent. $$ \det(A) = \det(A^T) $$

7. The dot product of two vectors can be calculated through the multiplication of the transpose of the first vector by the second vector. $$ \vec{a} \cdot \vec{b} = \vec{a}^T \vec{b} $$

8. For a matrix A composed solely of real entries, the product of the matrix by its transpose AA^T yields a positive semidefinite symmetric matrix.

9. An invertible matrix remains so when transposed, and its inverse is simply the transpose of the inverse of the original matrix. $$ (A^T)^{-1} = (A^{-1})^T $$

10. A matrix that satisfies the condition A^T=A^-1, where the transpose equals its inverse, earns the special distinction of being orthogonal.

11. For a square matrix A, its eigenvalues retain their identity, matching those of its transpose.