Linear Algebra/Addition, Multiplication, and Transpose

Addition and subtraction

Two matrices can only be added or subtracted if they have the same size. Matrix addition and subtraction are done entry-wise, which means that each entry in A+B is the sum of the corresponding entries in A and B.

Here is an example of matrix addition

$A=\left[\begin{array}{ccccc}7& & 5& & 3\\ 4& & 0& & 5\end{array}\right]B=\left[\begin{array}{ccccc}1& & 1& & 1\\ -1& & 3& & 2\end{array}\right]$

$A+B=\left[\begin{array}{ccccc}7+1& & 5+1& & 3+1\\ 4-1& & 0+3& & 5+2\end{array}\right]=\left[\begin{array}{ccccc}8& & 6& & 4\\ 3& & 3& & 7\end{array}\right]$

And an example of subtraction

$A=\left[\begin{array}{ccccc}7& & 5& & 3\\ 4& & 0& & 5\end{array}\right]B=\left[\begin{array}{ccccc}1& & 1& & 1\\ -1& & 3& & 2\end{array}\right]$

$A-B=\left[\begin{array}{ccccc}7-1& & 5-1& & 3-1\\ 4+1& & 0-3& & 5-2\end{array}\right]=\left[\begin{array}{ccccc}6& & 4& & 2\\ 5& & -3& & 4\end{array}\right]$

Remember you can not add or subtract two matrices of different sizes.

The following rules applies to sums and scalar multiples of matrices.
Let A, B, and C be matrices of the same size, and let r and s be scalars.

• A + B = B + A
• (A + B) + C = A + (B + C)
• A + 0 = A
• r(A + B) = rA + rB
• (r + s)A = rA + sA
• r(sA) = (rs)A

Powers

If A is an $n\times n$ matrix and if k is a positive integer, then $A^k$ denotes the product of k copies of A

$A^k=\begin{array}{c}\underbrace{A\cdots A}\\ k\end{array}$

If A is nonzero and if x is in ${ℝ}^n$, then $A^k𝐱$ is the result of left-multiplying x by A repeatedly k times. If k = 0, then $A^0𝐱$ should be x itself. Thus $A^0$ is interpreted as the identity matrix.

Transpose

Given the $m\times n$ matrix A, the transpose of A is the $n\times m$, denoted $A^T$, whose columns are formed from the corresponding rows of A.

For example

$A=\left[\begin{array}{ccc}a& & b\\ c& & d\end{array}\right]B=\left[\begin{array}{ccc}3& & 5\\ 2& & 7\\ 6& & 9\\ 1& & 0\\ 5& & 2\end{array}\right]$

$A^T=\left[\begin{array}{ccc}a& & c\\ b& & d\end{array}\right]B^T=\left[\begin{array}{ccccccccc}3& & 2& & 6& & 1& & 5\\ 5& & 7& & 9& & 0& & 2\end{array}\right]$

The following rules applied when working with transposing

1. ${\left(A^T\right)}^T=A$
2. $(A+B{)}^T=A^T+B^T$
3. For any scalar r, $(rA{)}^T=r{A}^T$
4. $(AB{)}^T=B^T{A}^T$

The 4th rule can be generalize to products of more than two factors, as "The transpose of a product of matrices equals the product of their transposes in the reverse order." Meaning

<math>(a_1, a_2, a_3, ..., a_n)^T=a_n^T, ..., a_3^T, a_2^T, a_1^T