Diagonal Matrices

\( \) \( \) \( \)

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.
\( A = \begin{bmatrix} -6 & 0 \\ 0 & 9 \end{bmatrix} \)     \( B = \begin{bmatrix} 9 & 0 & 0\\ 0 & 7 & 0\\ 0 & 0 & -1 \end{bmatrix} \)     \( C = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 7 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -5 \end{bmatrix} \)     \( D = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \)



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 \( A = \begin{bmatrix} a_{11} & 0 & 0 & ... \\ 0 & a_{22} & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & a_{nn} \end{bmatrix} \) and \( B = \begin{bmatrix} b_{11} & 0 & 0 & ... \\ 0 & b_{22} & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & b_{nn} \end{bmatrix} \)

Addition: \( A + B = \begin{bmatrix} a_{11} + b_{11} & 0 & 0 & ... \\ 0 & a_{22} + b_{22} & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & a_{nn} + b_{nn} \end{bmatrix} \) is a diagonal matrix.

Mutliplication: \( A B = \begin{bmatrix} a_{11} b_{11} & 0 & 0 & ... \\ 0 & a_{22} b_{22} & 0 & ... \\ . & & & \\ . & & & \\ . & & & \\ 0 & 0 & ... & a_{nn} b_{nn} \end{bmatrix} \) is a diagonal matrix.



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
\( \lambda_1 = a_{11}\)   ,   \( e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ . \\ . \\ 0 \end{bmatrix} \)

\( \lambda_2 = a_{22}\)   ,   \( e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \\ . \\ . \\ 0 \end{bmatrix} \)
...
etc
...
\( \lambda_n = a_{nn}\)   ,   \( e_{n} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ . \\ . \\ 1 \end{bmatrix} \)



Examples with Solutions

Example 1
Which of the following 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) \( D = \begin{bmatrix} 5 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 9 & 0\\ 0 & 0 & 0 & -1 \end{bmatrix} \)

Solution
Matrices \( B \) and \( D \) are diagonal matrices.


Example 2
Matrices \( A \) and \( B \) are given by \( A = \begin{bmatrix} -1 & 0 & 0\\ 0 & -\dfrac{2}{3} & 0 \\ 0 & 0 & \dfrac{1}{4} \end{bmatrix} \) , \( B = \begin{bmatrix} -1 & 0 & 0\\ 0 & -6 & 0 \\ 0 & 0 & 8 \end{bmatrix} \).
Calculate \( A B \)

Solution
\( A B = \begin{bmatrix} -1 & 0 & 0\\ 0 & -\dfrac{2}{3} & 0 \\ 0 & 0 & \dfrac{1}{4} \end{bmatrix} \begin{bmatrix} -1 & 0 & 0\\ 0 & -6 & 0 \\ 0 & 0 & 8 \end{bmatrix} \)

\( \quad \quad = \begin{bmatrix} (-1)(-1) & 0 & 0\\ 0 & (-\dfrac{2}{3})(-6) & 0 \\ 0 & 0 & (\dfrac{1}{4})(8) \end{bmatrix} \)

\( \quad \quad = \begin{bmatrix} 1 & 0 & 0\\ 0 & 4 & 0 \\ 0 & 0 & 2 \end{bmatrix} \)



Example 3
Let matrix \( A = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & - 2 \end{bmatrix} \)
Calculate \( A^5 \).

Solution
\( A^5 = \begin{bmatrix} (-1)^5 & 0 & 0 & 0 \\ 0 & (2)^5 & 0 & 0 \\ 0 & 0 & (-1)^5 & 0 \\ 0 & 0 & 0 & (-2)^5 \end{bmatrix} \)
Simplify
\( \quad \quad = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 32 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -32 \end{bmatrix} \)



Example 4
Find the eigenvalues and eigenvectors of the matrix
\( A = \begin{bmatrix} 1 & 0\\ 0 & -3 \end{bmatrix} \)

Solution
Eigenvalue: \( \lambda_1 = 1 \)   ,   Eigenvector: \( e_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \)

Eigenvalue: \( \lambda_2 = -3 \)   ,   Eigenvector: \( e_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \)



Example 5
Find the eigenvalues and eigenvectors of the matrix
\( A = \begin{bmatrix} -6 & 0 & 0\\ 0 & \dfrac{1}{5} & 0 \\ 0 & 0 & - \dfrac{2}{3} \end{bmatrix} \)

Solution
Eigenvalue: \( \lambda_1 = -6 \)   ,   Eigenvector: \( e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \)

Eigenvalue: \( \lambda_2 = \dfrac{1}{5} \)   ,   Eigenvector: \( e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \)

Eigenvalue: \( \lambda_3 = - \dfrac{2}{3} \)   ,   Eigenvector: \( e_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \)



Example 6
Find the eigenvalues and eigenvectors of the matrix
\( A = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 9 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix} \)

Solution
Eigenvalue: \( \lambda_1 = 1 \)   ,   Eigenvector: \( e_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} \)

Eigenvalue: \( \lambda_2 = -1 \)   ,   Eigenvector: \( e_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} \)

Eigenvalue: \( \lambda_3 = 9 \)   ,   Eigenvector: \( e_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} \)

Eigenvalue: \( \lambda_4 = 1 \)   ,   Eigenvector: \( e_4 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \)



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: \( e_4 = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \)

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