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.
Diagonal Matrices


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 n given by

Matrix A    and    Matrix B

Addition: Add Diagonal Matrix A+B is a diagonal matrix.

Multiplication: Mutliply Diagonal Matrix A+B 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.



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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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