Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Symmetric matrices

How to show that:

A A^T = \displaystyle \sum_i a_i a_i^T

, where

A

is symmetric matrix?

Example:

When

A=\left[\begin{array}{ccc}1&2&3\\2&3&1\\3&1&2\end{array}\right]

then

A A^T = \left[ \begin{array}{ccc} 1\\2\\3 \end{array} \right] \left[ \begin{array}{ccc} 1&2&3 \end{array} \right] + \left[ \begin{array}{ccc} 2\\3\\1 \end{array} \right] \left[ \begin{array}{ccc} 2&3&1 \end{array} \right] + \left[ \begin{array}{ccc} 3\\1\\2 \end{array} \right] \left[ \begin{array}{ccc} 3&1&2 \end{array} \right]

So

a_i

is the column vector (i-th column).

If

A

is symmetric then

A = A^T

and

A

is an

n \times n

matrix.

AA^T = (A_{ij})^2 = \displaystyle\sum_{k=1}^n a_{ik}a_{kj}

From the definition of matrix multiplication. Do you see how to finish it?

(edited 11 years ago)

OP

Thank you for reply.

\displaystyle\sum_{k=1}^n a_{ik}a_{kj} = \displaystyle\sum_{k=1}^n a_{ik}a_{jk}^T

So now can I rewrite as:

\displaystyle\sum_{k=1}^n a_k a_k^T

or would it be incorrect?

Original post by Noble.

If

A

is symmetric then

A = A^T

and

A

is an

n \times n

matrix.

AA^T = (A_{ij})^2 = \displaystyle\sum_{k=1}^n a_{ik}a_{kj}

From the definition of matrix multiplication. Do you see how to finish it?

I am being a pedant here but you should really write something like

\left(\displaystyle\sum_{k=1}^n a_{ik}a_{kj}\right)

since the inside of the bracket is just a component; not a matrix.

Original post by Mark85

I am being a pedant here but you should really write something like

\left(\displaystyle\sum_{k=1}^n a_{ik}a_{kj}\right)

since the inside of the bracket is just a component; not a matrix.

Interesting. They don't do this in the lecture notes.

Original post by Noble.

Interesting. They don't do this in the lecture notes.

Well, you wrote brackets round the

A_{ij}

which is just the i,jth component of A. It doesn't matter whether you choose to use brackets around a typical component to signify you mean the whole matrix or not but you should at least be consistent.

I mean, sure it is very clear what is meant but such things might cause a problem for people learning e.g. the OP.

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