Diagonal Matrices

Page Content

• Definition of a Matrix
• Properties of the Diagonal Matrices
• Addition and Multiplication of Diagonal Matrices
• Inverse of Diagonal Matrices
• Power of Diagonal Matrices
• Examples with Solutions
• Questions (with solutions given below)
• References

Definition of a Diagonal Matrix

A diagonal matrix is a square matrix in which all entries off the main diagonal are zero.
These are examples of diagonal matrices.
\[ A = \begin{bmatrix} -6 & 0 \\ 0 & 9 \end{bmatrix} \qquad B = \begin{bmatrix} 9 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & -1 \end{bmatrix} \] \[ C = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 7 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -5 \end{bmatrix} \qquad D = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \]

Properties of the Diagonal Matrices

Some of the most important properties of the diagonal matrices are given below.

1.    The determinant of a diagonal matrix is equal to the product of all entries in the main diagonal.
2.    A diagonal matrix is a symmetric matrix.
3.    A diagonal matrix is invertible (it has an inverse) if and only if none of its entries in the main diagonal is zero.

Addition and Multiplication of Diagonal Matrices

Let A and B be two diagonal matrices of the same size \( n \times n \) given by \[ A = \begin{bmatrix} a_{11} & 0 & 0 & \cdots \\ 0 & a_{22} & 0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ 0 & 0 & \cdots & a_{nn} \end{bmatrix} \] and \[ B = \begin{bmatrix} b_{11} & 0 & 0 & \cdots \\ 0 & b_{22} & 0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ 0 & 0 & \cdots & b_{nn} \end{bmatrix} \]
Addition: \[ A + B = \begin{bmatrix} a_{11} + b_{11} & 0 & 0 & \cdots \\ 0 & a_{22} + b_{22} & 0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ 0 & 0 & \cdots & a_{nn} + b_{nn} \end{bmatrix} \] Multiplication: \[ AB = \begin{bmatrix} a_{11} b_{11} & 0 & 0 & \cdots \\ 0 & a_{22} b_{22} & 0 & \cdots \\ \vdots & \vdots & \vdots & \ddots \\ 0 & 0 & \cdots & a_{nn} b_{nn} \end{bmatrix} \]

Inverse of Diagonal Matrices

Let \( A \) be a diagonal matrix given by \( A = \begin{bmatrix} a_{11} & 0 & 0 & ... \\ 0 & a_{22} & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & a_{nn} \end{bmatrix} \)
If none of the entries \( a_{ii} \) in the main diagonal is equal to zero, then \( A \) is invertible and its inverse is given by \[ A^{-1} = \begin{bmatrix} \dfrac{1}{a_{11}} & 0 & 0 & ... \\ 0 & \dfrac{1}{a_{22}} & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & \dfrac{1}{a_{nn}} \end{bmatrix} \]

Power of Diagonal Matrices

Let \( A \) be a diagonal matrix given by \( A = \begin{bmatrix} a_{11} & 0 & 0 & ... \\ 0 & a_{22} & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & a_{nn} \end{bmatrix} \)
\[ A^p = \begin{bmatrix} a_{11}^p & 0 & 0 & ... \\ 0 & a_{22}^p & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & a_{nn}^p \end{bmatrix} \] where \( p \) is a positive integer.

Eigenvalues and Eigenvectors of Diagonal Matrices

Let \( A \) be a diagonal matrix given by \[ A = \begin{bmatrix} a_{11} & 0 & 0 & ... \\ 0 & a_{22} & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & a_{nn} \end{bmatrix} \] The eigenvalues \( \lambda_i \) and corresponding eigenvectors \( e_i \) are given by \[ \lambda_1 = a_{11} \; , \quad e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ . \\ . \\ 0 \end{bmatrix} \] \[ \lambda_2 = a_{22} \; , \quad e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \\ . \\ . \\ 0 \end{bmatrix} \] ... etc ... \[ \lambda_n = a_{nn} \; , \quad e_{n} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ . \\ . \\ 1 \end{bmatrix} \]

Examples with Solutions

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Questions (with solutions given below)

• Part 1
  Find the real constants \( a \) and \( b \) such that \( A^{-1} = A \) where \( A \) is a matrix given by \( A = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & \dfrac{1}{a} & 0 & 0\\ 0 & 0 & b+1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix} \)
  
• Part 2
  Find the eigenvalues and eigenvectors of the matrix
  \( A = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} \)
  
• Part 3
  Find the eigenvalues and eigenvectors of the matrix
  \( A = \begin{bmatrix} 2 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & -2 & 0\\ 0 & 0 & 0 & -2 \end{bmatrix} \)

Solutions to the Above Questions

• Part 1
  We first find the inverse of the given matrix \( A^{-1} = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & a & 0 & 0\\ 0 & 0 & \dfrac{1}{b+1} & 0\\ 0 & 0 & 0 & 1 \end{bmatrix} \)
  In order to have \( A^{-1} = A \), we need to have
  \( a = \dfrac{1}{a} \)    (I) and \( \dfrac{1}{b+1} = b + 1 \)    (II)
  Solve the above equations. Equation (I) may be written as
  \( a^2 = 1 \)
  and gives two solutions: \( a = 1 \) and \( a = - 1\)
  Equation (II) may be written as
  \( (b+1)^2 = 1 \)
  and gives two solutions: \( b = 0 \) and \( b = - 2 \)
  Hence the solutions to the given questions are the following pairs:
  \( a = 1 \) , \( b = 0 \)
  \( a = 1 \) , \( b = -2 \)
  \( a = - 1 \) , \( b = 0 \)
  \( a = - 1 \) , \( b = -2 \)
  
  
• Part 2
  
  Eigenvalue: \( \lambda_1 = 1 \)   ,   Eigenvector: \( e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \)
  Eigenvalue: \( \lambda_2 = 1 \)   ,   Eigenvector: \( e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \)
  Eigenvalue: \( \lambda_3 = 0 \)   ,   Eigenvector: \( e_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \)
  
  
• Part 3
  Eigenvalue: \( \lambda_1 = 2 \)   ,   Eigenvector: \( e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} \)
  Eigenvalue: \( \lambda_2 = 2 \)   ,   Eigenvector: \( e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} \)
  Eigenvalue: \( \lambda_3 = -2 \)   ,   Eigenvector: \( e_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} \)
  Eigenvalue: \( \lambda_4 = -2 \)   ,   Eigenvector: \( e_4 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \)

More References and links

• Matrices with Examples and Questions with Solutions.
• Determinant of a Square Matrix.
• Eigenvalues and Eigenvectors.
• Inverse Matrix Questions with Solutions.
• Elementary Linear Algebra - 7 th Edition - Howard Anton and Chris Rorres
• Introduction to Linear Algebra - Fifth Edition (2016) - Gilbert Strang
• Linear Algebra Done Right - third edition, 2015 - Sheldon Axler
• Linear Algebra with Applications - 2012 - Gareth Williams