Matrix transposition: continuing learning by doing
Definition. For any matrix its transposed is obtained by putting rows of into columns of .
Exercise 1. See Figure 1 borrowed from Wikipedia. Based on that illustration, what is the relationship between and
Exercise 2. For the matrix which of the products and exists? Find the one(s) that exist. Repeat the same for
Definition. If then is called symmetric.
Exercise 3. Can a non-square matrix be symmetric? The above definition is in matrix terms. What does it mean in terms of matrix elements?
Exercise 4. What is the relationship between and Consider just a matrix.
Exercise 5. What happens if you apply transposition to a product ?
Solution. Partitioning and into rows and columns, respectively, we find the elements of the product as dot products of the rows of by the columns of :
In the second expression the dots are omitted because the are rows and the are columns, so that the matrix product definition can be applied to write Further, are rows and are columns, so we can write
Transposing (1) and using (2) we have
We have proved that
Exercise 6. What is the relationship between transposition and inversion? More precisely, if is invertible, then what can you say about ?
Solution. By definition, Transposing this gives This shows that