Diagonal Matrix Examples

What is Diagonal Matrix?

In mathematics, the term diagonals matrix define as the matrix in which the off diagonals entries are zero and main diagonals entries are some else. There is no restriction for main diagonals entries. diagonals matrix is also called the scaling matrix because when we multiply any matrix with diagonals matrix it change the scale of that matrix here the scale meaning size of the matrix. The identity matrix is one of the examples of the diagonals matrix. because in the identity matrix the main diagonals are 1 and off diagonals are zero.

Diagonal Matrix Definition?

Diagonals matrix must be a square matrix. Since the square matrix is a symmetric matrix so we also called this matrix the symmetric diagonals matrix.

Examples of diagonals matrix

Example no 1:
\begin{vmatrix}6&0\\0&4\end{vmatrix}

It is a 2 by 2 diagonals matrix.

Example no 2:
\;\begin{bmatrix}7&0&0\\0&5&0\\0&0&2\end{bmatrix}

It is 3 by 3 diagonals matrix.

Example no 3:
\begin{bmatrix}7&0&0&0\\0&5&0&0\\0&0&4&0\\0&0&0&9\end{bmatrix}

it is a 4 by 4 diagonals matrix.

Properties of Diagonals matrix

  • When we add two diagonals matrices then the resultant matrix also diagonals matrix. \;\begin{bmatrix}1&0&0\\0&4&0\\0&0&3\end{bmatrix}\;\;+\begin{bmatrix}7&0&0\\0&5&0\\0&0&2\end{bmatrix}\;=\begin{bmatrix}8&0&0\\0&9&0\\0&0&5\end{bmatrix}
  • If any matrix is triangular and normal then then it is diagonal matrix.
  • both the upper and lower triangular matrix are diagonals matrix.
  • Symmetric matrix also diagonals matrix.
  • When we multiply any matric with diagonals matric then the matrix which we obtain is diagonals matrix \;\begin{bmatrix}2&0&0\\0&4&0\\0&0&5\end{bmatrix}\;\;\times\;\;\begin{bmatrix}4&0&0\\0&3&0\\0&0&2\end{bmatrix}\;=\begin{bmatrix}8&0&0\\0&12&0\\0&0&10\end{bmatrix}
  • Zero matrix is diagonals matrix. \;\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}\; it is a 3 by 3 zero matrix and it hold the property of diagonal matrix.
  • Identity matrix also diagonals matrix \;\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix} this is the 3 by 3 identity matrix since it is hold the property of diagonals matrix
  • The matrix of order 1 by 1 is diagonals matrix.
  • Transpose of any diagonals matrix is same as matrix.
  • Inverse of diagonals matrix is resiprocal of main diagonals entries .
  • in under multiplication the diagonals matrix are commutative as AB = BA

Note Diagonal Matrix

  • since 1 by 1 order matrix is also diagonal matrix.
  • there is no restrication for diagonal entries.

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