Matrix transposition: continuing learning by doing
Definition. For any matrix its transposed
is obtained by putting rows of
into columns of
.

Figure 1. Matrix transpose
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
:
(1)
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
(2)
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
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