A diagonal matrix is a square matrix in which all entries off the main diagonal are zero .
Below are examples of diagonal matrices of various dimensions :
$$ A = \begin{bmatrix} -6 & 0 \\ 0 & 9 \end{bmatrix} \qquad B = \begin{bmatrix} 9 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & -1 \end{bmatrix} $$ $$ C = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 7 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -5 \end{bmatrix} \qquad D = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$Some of the most important properties of diagonal matrices include :
Let \( A \) and \( B \) be two diagonal matrices of the same size \( n \times n \) :
$$ A = \begin{bmatrix} a_{11} & 0 & 0 & \cdots \\ 0 & a_{22} & 0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ 0 & 0 & \cdots & a_{nn} \end{bmatrix}, \quad B = \begin{bmatrix} b_{11} & 0 & 0 & \cdots \\ 0 & b_{22} & 0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ 0 & 0 & \cdots & b_{nn} \end{bmatrix} $$Matrix Addition:
$$ A + B = \begin{bmatrix} a_{11} + b_{11} & 0 & 0 & \cdots \\ 0 & a_{22} + b_{22} & 0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ 0 & 0 & \cdots & a_{nn} + b_{nn} \end{bmatrix} $$Matrix Multiplication:
$$ AB = \begin{bmatrix} a_{11} b_{11} & 0 & 0 & \cdots \\ 0 & a_{22} b_{22} & 0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ 0 & 0 & \cdots & a_{nn} b_{nn} \end{bmatrix} $$Let \( A \) be a diagonal matrix given by :
$$ A = \begin{bmatrix} a_{11} & 0 & 0 & \cdots \\ 0 & a_{22} & 0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ 0 & 0 & \cdots & a_{nn} \end{bmatrix} $$If none of the entries \( a_{ii} \) on the main diagonal equal zero, then \( A \) is invertible, and its inverse is :
$$ A^{-1} = \begin{bmatrix} \dfrac{1}{a_{11}} & 0 & 0 & \cdots \\ 0 & \dfrac{1}{a_{22}} & 0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ 0 & 0 & \cdots & \dfrac{1}{a_{nn}} \end{bmatrix} $$For any positive integer \( p \), the power of a diagonal matrix \( A \) is computed by raising each diagonal entry to the power \( p \) :
$$ A^p = \begin{bmatrix} a_{11}^p & 0 & 0 & \cdots \\ 0 & a_{22}^p & 0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ 0 & 0 & \cdots & a_{nn}^p \end{bmatrix} $$For a diagonal matrix \( A \), the eigenvalues \( \lambda_i \) are simply the entries on the main diagonal, and their corresponding standard unit vectors form the eigenvectors \( e_i \) :
$$ \lambda_1 = a_{11}, \quad e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \qquad \lambda_2 = a_{22}, \quad e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \qquad \dots \qquad \lambda_n = a_{nn}, \quad e_n = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \end{bmatrix} $$Which of the following matrices are diagonal matrices ?
a) \( A = \begin{bmatrix} 0 & 0 \\ 1 & 2 \end{bmatrix} \)
b) \( B = \begin{bmatrix} -2 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)
c) \( C = \begin{bmatrix} 5 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 5 & 0 & 0 & 4 \end{bmatrix} \)
d) \( D = \begin{bmatrix} 5 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 9 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix} \)
Matrices \( B \) and \( D \) are diagonal matrices because all entries off their main diagonals are strictly zero .
Given matrices \( A = \begin{bmatrix} -1 & 0 & 0 \\ 0 & -\dfrac{2}{3} & 0 \\ 0 & 0 & \dfrac{1}{4} \end{bmatrix} \) and \( B = \begin{bmatrix} -1 & 0 & 0 \\ 0 & -6 & 0 \\ 0 & 0 & 8 \end{bmatrix} \), calculate the product \( AB \) .
Let matrix \( A = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -2 \end{bmatrix} \). Calculate \( A^5 \) .
Find the eigenvalues and eigenvectors of matrix \( A = \begin{bmatrix} 1 & 0 \\ 0 & -3 \end{bmatrix} \) .
First Eigenpair: \( \lambda_1 = 1 \), \( e_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \)
Second Eigenpair: \( \lambda_2 = -3 \), \( e_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \)
Find the eigenvalues and eigenvectors of matrix \( A = \begin{bmatrix} -6 & 0 & 0 \\ 0 & \dfrac{1}{5} & 0 \\ 0 & 0 & -\dfrac{2}{3} \end{bmatrix} \) .
\( \lambda_1 = -6 \), \( e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \)
\( \lambda_2 = \dfrac{1}{5} \), \( e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \)
\( \lambda_3 = -\dfrac{2}{3} \), \( e_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \)
Find the eigenvalues and eigenvectors of matrix \( A = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 9 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \) .
\( \lambda_1 = 1 \), \( e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} \)
\( \lambda_2 = -1 \), \( e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} \)
\( \lambda_3 = 9 \), \( e_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} \)
\( \lambda_4 = 1 \), \( e_4 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \)
Part 1: Find the real constants \( a \) and \( b \) such that \( A^{-1} = A \) where \( A = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \dfrac{1}{a} & 0 & 0 \\ 0 & 0 & b+1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \) .
Part 2: Find the eigenvalues and eigenvectors of matrix \( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} \) .
Part 3: Find the eigenvalues and eigenvectors of matrix \( A = \begin{bmatrix} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & -2 & 0 \\ 0 & 0 & 0 & -2 \end{bmatrix} \) .
Solution to Part 1:
First, find the inverse matrix :
$$ A^{-1} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & a & 0 & 0 \\ 0 & 0 & \dfrac{1}{b+1} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$Setting \( A^{-1} = A \) gives equations for diagonal entries :
\( a = \dfrac{1}{a} \implies a^2 = 1 \implies a = 1 \) or \( a = -1 \)
\( \dfrac{1}{b+1} = b+1 \implies (b+1)^2 = 1 \implies b = 0 \) or \( b = -2 \)
Thus, valid pairs are :
\( (a=1, b=0), \quad (a=1, b=-2), \quad (a=-1, b=0), \quad (a=-1, b=-2) \)
Solution to Part 2:
\( \lambda_1 = 1 \), \( e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \)
\( \lambda_2 = 1 \), \( e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \)
\( \lambda_3 = 0 \), \( e_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \)
Solution to Part 3:
\( \lambda_1 = 2 \), \( e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} \)
\( \lambda_2 = 2 \), \( e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} \)
\( \lambda_3 = -2 \), \( e_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} \)
\( \lambda_4 = -2 \), \( e_4 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \)