# permutation matrix question

• Mar 6th 2011, 11:31 AM
situation
permutation matrix question
the question highlighted below is in two parts, really have no clue how to start it, any help would be greatly appreciated!!

a. Recall that an elementary permutation matrix is an n x n matrix which is In except that two rows of
In have been swapped. If $\displaystyle P_{i;j}$ is the elementary permutation matrix where rows i and j have been swapped
and $\displaystyle A$ is a matrix (with n rows), describe the relationship between $\displaystyle A$ and $\displaystyle P_{i;j}A$.

b. Let A be the matrix:

$\displaystyle A=\left[\begin{array}{ccc}1&2&3\\2&4&6\\1&3&5\end{array}\r ight]$

Find a permutation matrix P, a lower triangular matrix L and an upper triangular matrix U so that
$\displaystyle A = P^{T}LU$:

thank you!
• Mar 6th 2011, 12:29 PM
HallsofIvy
Quote:

Originally Posted by situation
the question highlighted below is in two parts, really have no clue how to start it, any help would be greatly appreciated!!

a. Recall that an elementary permutation matrix is an n x n matrix which is In except that two rows of
In have been swapped. If $\displaystyle P_{i;j}$ is the elementary permutation matrix where rows i and j have been swapped
and $\displaystyle A$ is a matrix (with n rows), describe the relationship between $\displaystyle A$ and $\displaystyle P_{i;j}A$.

Have you looked at an example? Write down any permutation matrix, P, any matrix A, and multiply them. Compare A and PA. The answer should be obvious.

Quote:

b. Let A be the matrix:

$\displaystyle A=\left[\begin{array}{ccc}1&2&3\\2&4&6\\1&3&5\end{array}\r ight]$

Find a permutation matrix P, a lower triangular matrix L and an upper triangular matrix U so that
$\displaystyle A = P^{T}LU$:

thank you!
• Mar 6th 2011, 01:36 PM
situation
Quote:

Originally Posted by HallsofIvy
Have you looked at an example? Write down any permutation matrix, P, any matrix A, and multiply them. Compare A and PA. The answer should be obvious.

ok so am i right in saying that if i took any identity matrix, swapped any of the two rows, that would be denoted a permutation matrix?

eg. $\displaystyle \left(\begin{array}{ccc}1&0&0\\0&0&1\\0&1&0\end{ar ray}\right)$ would be a permutation matrix on the identity matrix for a 3x3??
• Mar 7th 2011, 01:59 AM
Ackbeet
You're correct. However, your problem has asked you to compare the result when you take the matrix you exhibited in your last post and left-multiply it with A, versus just plain A itself. That is, if P is the matrix in your last post, then what's the difference between A and PA?