1.4 Basic Matrix Operations

We assume the usual matrix operations of addition, subtraction, scalar multiplication, inner product, multiplication, transpose, trace, determinant and inverse.

Example 1.5.

If \(A = \begin {pmatrix} 1 & 3 & 2\\ -1 & 4 & 0\\ 2 & 5 & 2\\ \end {pmatrix}\)

1.

\(\displaystyle {Tr(A) = \sum ^3_{i=1}a_{ii} = a_{11} + a_{22} + a_{33} = 1 + 4 + 2 = 7}\)

2.

\begin {align*} \det (A) & = |A| = 1 \begin {vmatrix} 4 & 0\\ 5 & 2\\ \end {vmatrix} - (-1) \begin {vmatrix} 3 & 2\\ 5 & 2\\ \end {vmatrix} + 2 \begin {vmatrix} 3 & 2\\ 4 & 0\\ \end {vmatrix}\\\\ & = (8-0) + (6-10) + 2(0-8)\\ & = 8 - 4 - 16\\ & = -12\\ \end {align*}

3.

\(\displaystyle { B = A + A' = \begin {pmatrix} 1 & 3 & 2\\ -1& 4 & 0\\ 2 & 5 & 2\\ \end {pmatrix} + \begin {pmatrix} 1 & -1 & 2\\ 3 & 4 & 5\\ 2 & 0 & 0\\ \end {pmatrix} = \begin {pmatrix} 2 & 2 & -4\\ 2 & 8 & 5\\ 4 & 5 & 4\\ \end {pmatrix} = B'}\)

Note 1.6.

(a): \(B = (b_{ij})\)
(b): \(B' = (b_{ji})\)
(c): \(B = B'\)

1.4.1 The Trace Function

The trace function satisfies the following properties: \[A_{p\times p}, B_{p\times p}, C_{p\times n}, D_{n\times p}\qquad \text {and scalar}\quad \alpha \]

1.: \(Trace(\alpha ) = \alpha \)
2.: \(Tr(A\pm B) = Tr(A) \pm Tr(B)\)
3.: \(Tr(\alpha A) = \alpha Tr(A)\)
4.: \(Tr(CD) = Tr(DC) = \displaystyle {\sum c_{ij}d_{ji}}\)

1.4.2 Square Matrix

A square matrix is non-singular if \(|A| \neq 0\) , otherwise it is singular.

The following results follow

1.: \(|A|= \displaystyle {\sum ^p_{j=1}a_{ij}A_{ij}}\), where \(A_{ij}\) is the determinant of the cofactor \(\displaystyle {A_{ij}=(-1)^{i+j}|A_{ij}|}\).
2.: If \(A\) is triangular or diagonal then \(\displaystyle {|A| = \prod ^p_{i=1} a_{ii}}\).
3.: \(\displaystyle {|\alpha A| = \alpha ^p |A|}\)
4.: \(\displaystyle {|AB| = |A||B|}\)
5.: For square sub-matrices \(A_{p\times p}\) and \(B_{q\times q}\) \[ \begin {vmatrix} A & C\\ 0 & B\\ \end {vmatrix} = \begin {vmatrix} A\\ \end {vmatrix}\begin {vmatrix} B\\ \end {vmatrix} \]
\[A^* = \begin {pmatrix} A & C\\ 0 & B\\ \end {pmatrix}_{(p + q)\times (p + q)} \]
6.: \(\displaystyle { \begin {vmatrix} A_{11} & A_{12}\\ A_{21} & A_{22}\\ \end {vmatrix}=\begin {vmatrix} A_{11}\\ \end {vmatrix}\begin {vmatrix} A_{22} - A_{21}A^{-1}_{11}A_{12}\\ \end {vmatrix} }\) for square \(A_{11}\) and \(A_{22}\).

Proof. Assume \(A^{-1}_{11}\) exists \( \begin {pmatrix} A_{11} & A_{12}\\ \end {pmatrix} \). Pre-multiply “row 1” by \(A_{21}A^{-1}_{11}\) and subtract from
“row 2” □

\begin {align*} \begin {vmatrix} A_{11} & A_{12}\\ A_{21}-A_{21}A_{11}^{-1}A_{11} & A_{22}-A_{21}A^{-1}_{11}A_{12}\\ \end {vmatrix} & = \begin {vmatrix} A_{11} & A_{12}\\ 0 & A_{22} - A_{21}A_{11}^{-1}A_{12}\\ \end {vmatrix}\\\\ & = \begin {vmatrix} A_{11}\\ \end {vmatrix}\begin {vmatrix} A_{22} -A_{21}A_{11}^{-1}A_{12}\\ \end {vmatrix}\qquad \text {using result (5)}\\\\ \end {align*}

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