Matrix Revisions

Basic Definition and notations:

1. An \(m\times n\) matrix \(A\) is a rectangular array of numbers with \(m\) rows and \(n\) columns.

2. The elements of a matrix \(A_{m\times n}\) are \(a_{ij}\)

3. The order of a matrix is the number of rows by the number of columns, i.e. \(m\times n\)

4. A column vector with \(m\) elements, \(y = \left( \begin{array}{c} y_1\\ y_2\\ \vdots\\ y_n \end{array} \right)\), is a matrix with only one column i.e. an \(m\times 1\) matrix.

5. A row vector with \(n\) elements, \(x=(x_1,x_2,x_3, \cdots, x_n)\), is a matrix with only one row, i.e. an \(1\times n\) matrix.

6. Transposed matrix \(A^T\) (or \(A'\)) arises from the matrix \(A\) by interchanging the column vectors and the row vectors i.e. \(a_{ij}^T = a_{ji}\) (so a column vector is converted into a row vector and vise versa)

7. A partitioned matrix is a matrix written in terms of sub-matrices. \(A=\left( \begin{array}{c} A_{11} & A_{12}\\ A_{21} & A_{22}\\ \end{array} \right)\), where \(A_{11},A_{12},A_{21},A_{22}\) are sub-matrices

   1. \(A_{11}, A_{21}\) have the same number of columns, so do \(A_{12}, A_{22}\)
   2. \(A_{11}, A_{12}\) have the same number of rows, so do \(A_{21}, A_{22}\)
   3. partitioning is not restricted to dividing a matrix into just four sub-matrices

8. A square matrix has exactly as many rows as it has columns i.e. the order of the matrix is \(n\times n\)

9. The main diagonal (or leading diagnonal) of a square matrix \(A (n\times n)\) are the elements lying on the diagnoal from top left to bottom right. \(a_{11},a_{22},a_{33},\cdots,a_{nn}\) i.e. all \(a_{ii}, i= 1,\cdots, n\)

10. The trace of a square matrix is the sum of the diagonal elements \(tr(A)=a_{11}+a_{22}+\cdots+a_{nn}=\sum_{i=1}^na_{ii}\)

Special matrices

1. A symmetric matrix is a square matrix for which the following is true for all the off diagonal elements. \(a_{ij}=a_{ji}\) i.e. \(A^T=A\)
2. Diagonal matrix is a square matrix having zero for all the non-diagonal elements i.e. \(A=\left( \begin{array}{c} a_{11} & \cdots & 0\\ \vdots & \ddots & \vdots\\ 0 & \cdots & a_{nn} \end{array} \right)\)
3. Zero matrix (null matrix) is a matrix whose all elements are zero
4. Identity matrix (or unit matrix) is a diagonal matrix having all diagonal elements equal to 1 and off diagonal elements equal to zero. i.e. \(I=\left( \begin{array}{c} 1 & \cdots & 0\\ \vdots & \ddots & \vdots\\ 0 & \cdots & 1 \end{array} \right)\)
5. “Summing vector” is a vector whose every element is 1 i.e. \(1_{n}=(1\cdots1)\)
6. “J matrix” is a matrix (not necessarily square) whose every element is 1 i.e. \(J_{m\times n}=\left( \begin{array}{c} 1 & 1 & \cdots & 1\\ 1 & 1 & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots\\ 1 & 1 & \cdots & 1 \end{array} \right)\)

Basic Operations

• \(A+B=B+A\)
• \((A+B)+C=A+(B+C)\)
• \(A+0=0+A=A\)
• \(A+(-A)=0\)
• \((A+B)^T=A^T+B^T\)

Multiplication by scalar: - \(cA=Ac\) - \(c(dA)=(cd)A\) - \((c\pm d)A=cA\pm dA\) - \(c(A\pm B)=cA \pm cB\)

Multiplication of an \(2\times2\) matrix by a column vector which has 2 rows yields a column vector with \(2\) rows. \[Ax=\left( \begin{array}{c} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{array} \right)\left( \begin{array}{c} x_{1}\\ x_{2}\\ \end{array} \right)=\left( \begin{array}{c} a_{11}x_1+a_{12}x_2\\ a_{21}x_1+a_{22}x_2\\ \end{array} \right)\]

Further definitions

1. The determinant of a second order square matrix is \(det(A)=|A|=\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}=a_{11}a_{22}-a_{12}a_{21}\)

2. The inverse of a matrix \(A\), \(A^{-1}\) if it exists, is a matrix whose product with \(A\) is the identity matrix i.e. \(AA^{-1}=A^{-1}A=I\). (Note: both \(A\) and \(A^{-1}\) have to be square) For second order matrices:\(A^{-1}=\frac{1}{det(A)}\left( \begin{array}{c} a_{22} & -a_{12}\\ -a_{21} & a_{11}\\ \end{array} \right)\)

3. Singular or non-invertible matrix: \(det(A)=0\)

4. Idempotent matrices(冪等矩陣) are square and the following is true: \(AA=A^2=A\)

5. Orthogonal matrices have the following property: \(AA^T=A^TA=I\)