Identity, Scalar and Diagonal Matrices

1. Identity Matrix: Any Square Matrix having Numerical Value 1 as it's Main Diagonal Elements and Numerical Value 0 as Other Elements is called an Identity Matrix. Identity Matrices are denoted by letter \(I\). Following are some examples of Identity Matrices of Order 2, 3 and 4
   
   \(\begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}\hspace{.5cm}\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\0 & 0 & 1 \end{bmatrix}\hspace{.5cm}\begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}\)
   
   Let's consider a \(M \times N\) Matrix \(A\) as given below
   
   \(A=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & ... & a_{nn}\end{bmatrix}\)   ...(1)
   
   Pre-Multiplying a \(M\times M\) Identity Matrix to Matrix \(A\) as given in equation (1) above (or Post Multiplying a \(N\times N\) Identity Matrix to Matrix \(A\)) gives back the same Matrix \(A\). That is,
   
   \(IA = \begin{bmatrix} 1 & 0 & ... & 0\\ 0 & 1 & ... & 0 \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & ... & 1_m\end{bmatrix} \begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix} =\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix}=A\)   ...(2)
   
   \(AI=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix} \begin{bmatrix} 1 & 0 & ... & 0\\ 0 & 1 & ... & 0 \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & ... & 1_n\end{bmatrix} =\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix}=A\)   ...(3)
2. Scalar Matrix: Any Square Matrix having Same Non-Zero Numerical Value as it's Main Diagonal Elements and Numerical Value 0 as Other Elements is called a Scalar Matrix. Scalar Matrices are denoted by letter \(S\). Following are some examples of Scalar Matrices
   
   \(\begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0\\0 & 0 & 2 \end{bmatrix}\hspace{.5cm} \begin{bmatrix} -1.5 & 0 \\ 0 & -1.5\end{bmatrix}\)
   
   Pre-Multiplying a \(M\times M\) Scalar Matrix having Non Zero Value \(k\) to Matrix \(A\) as given in equation (1) above (or Post Multiplying a \(N\times N\) Scalar Matrix having Non Zero Value \(k\) to Matrix \(A\)) is same as Multiplying Matrix \(A\) with the Scalar Value \(k\). That is,
   
   \(SA=\begin{bmatrix} k & 0 & ... & 0\\ 0 & k & ... & 0 \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & ... & k_m\end{bmatrix} \begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix} =\begin{bmatrix} ka_{11} & ka_{12} & ... & ka_{1n}\\ ka_{21} & ka_{22} & ... & ka_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ka_{m1} & ka_{m2} & ... & ka_{mn}\end{bmatrix}=kA\)   ...(4)
   
   \(AS=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix} \begin{bmatrix} k & 0 & ... & 0\\ 0 & k & ... & 0 \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & ... & k_n\end{bmatrix} =\begin{bmatrix} ka_{11} & ka_{12} & ... & ka_{1n}\\ ka_{21} & ka_{22} & ... & ka_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ka_{m1} & ka_{m2} & ... & ka_{mn}\end{bmatrix}=kA\)   ...(5)
3. Diagonal Matrix: Any Square Matrix having Atleast 1 Non Zero Numerical Value as it's Main Diagonal Elements and Numerical Value 0 as All Other Elements is called a Diagonal Matrix. Diagonal Matrices are denoted by letter \(D\). Following are some examples of Diagonal Matrices
   
   \(\begin{bmatrix} 1.6 & 0 & 0 \\ 0 & 20 & 0\\0 & 0 & -3 \end{bmatrix}\hspace{.5cm} \begin{bmatrix} -2.8 & 0 \\ 0 & 0.1\end{bmatrix}\)
   
   Diagonal Matrices can be used for Scaling Individual Rows and/or Individual Columns of any given Matrix with a Different Value .
   
   Pre-Multiplying any \(M \times M\) Diagonal Matrix \(D\) to Matrix \(A\) as given in equation (1) above Scales Rows of Matrix \(A\). That is,
   
   \(DA=\begin{bmatrix} k_{1} & 0 & ... & 0\\ 0 & k_{2} & ... & 0\\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & ... & k_{m}\end{bmatrix}\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix}=\begin{bmatrix} k_1a_{11} & k_1a_{12} & ... & k_1a_{1n}\\ k_2a_{21} & k_2a_{22} & ... & k_2a_{2n}\\ \vdots & \vdots & \ddots & \vdots \\k_ma_{m1} & k_ma_{m2} & ... & k_ma_{mn}\end{bmatrix}\)   ...(6)
   
   As given in equation (6) above, Any Row of the Matrix \(A\) gets Scaled by the Non Zero Value of Same Row of Matrix \(D\).
   
   Post-Multiplying any \(N \times N\) Diagonal Matrix \(D\) to Matrix \(A\) as given in equation (1) above Scales Columns of Matrix \(A\). That is,
   
   \(AD=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & ... & a_{mn}\end{bmatrix}\begin{bmatrix} k_{1} & 0 & ... & 0\\ 0 & k_{2} & ... & 0 \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & ... & k_{n}\end{bmatrix}=\begin{bmatrix} k_1a_{11} & k_2a_{12} & ... & k_na_{1n}\\ k_1a_{21} & k_2a_{22} & ... & k_na_{2n} \\ \vdots & \vdots & \ddots & \vdots \\k_1a_{m1} & k_2a_{m2} & ... & k_na_{mn}\end{bmatrix}\)   ...(7)
   
   As given in equation (7) above, Any Column of the Matrix \(A\) gets Scaled by the Non Zero Value of Same Column of Matrix \(D\).