11. Matrix Operations

Dated: 18-03-2025

\((i - j)th\) Element of a Matrix

\[\text{Column }j\]

\[ \text{Row }i \begin{bmatrix} a_{11} & \cdots & a_{1j} & \cdots & a_{1n} \\ \vdots & \quad & \vdots & \quad & \vdots \\ a_{i1} & \cdots & a_{ij} & \cdots & a_{in} \\ \vdots & \quad & \vdots & \quad & \vdots \\ a_{m1} & \cdots & a_{mj} & \cdots & a_{mn} \\ \end{bmatrix} = A \]

\[ \begin{array}{ccccc} \quad & \uparrow & \quad & & \uparrow & \quad & &\uparrow && \\ & a_1 & && a_j & & &a_n \end{array} \]

Diagonal Matrix

A diagonal matrix is a square matrix¹ whose non-diagonal entries¹ are zero.

\[ D = \begin{bmatrix} d_{11} & 0 & \cdots & 0 \\ 0 & d_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & d_{nn} \end{bmatrix} \]

The entries¹ \(a_{ij}\) where \(i = j\) form the main diagonal.

Theorem

Let \(A, B\) and \(C\) be matrices¹ of the same size and \(r\) and \(s\) be the scalars.

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

Row Column Rule for Computing \(AB\)

\[A(B(\vec x)) = AB (\vec x)\]

The left side transforms \(\vec x\) according to \(B\) and then according to \(A\).
Meanwhile, the right side is a composition of those transforms.

Properties of Product

\(A(BC) = (AB)C\)
\(A(B + C) = AB + AC\)
\((B + C)A = BA + CA\)
\(r(AB) = (rA)B = A(rB)\)
\(I_mA = A = AI_m\)

Warnings

In general, \(AB \ne BA\)
In general, if \(AC = BC\) then it is not necessarily the case that \(A = B\)
In general, if \(AB = 0\) then it is not necessarily the case that \(A = 0\) or \(B = 0\)

Power of a Matrix

\[A^k = \prod^k A = \underbrace{A \times A \times \cdots \times A}_{k}\]

\[A^0 = I\]

Theorems

\((A^t)^t = A\)
\((A + B)^t = A^t + B^t\)
\((rA)^t = r A^t\)
\((AB)^t = B^tA^t\)