1. What Is a Diagonal Matrix?
A diagonal matrix is a square matrix in which all elements outside the main diagonal are zero. Only the elements on the main diagonal may be non-zero.
The main diagonal consists of the entries:
\( a_{11}, a_{22}, a_{33}, \dots, a_{nn} \)
In general, a diagonal matrix looks like:
\( D = \begin{bmatrix} d_1 & 0 & 0 & \cdots & 0 \\ 0 & d_2 & 0 & \cdots & 0 \\ 0 & 0 & d_3 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & d_n \end{bmatrix} \)
2. Understanding the Structure of a Diagonal Matrix
The defining feature of a diagonal matrix is that every off-diagonal entry is zero. The diagonal entries can be zero or non-zero, but the rest must be zero.
2.1. Example of a Diagonal Matrix
\( D = \begin{bmatrix} 5 & 0 & 0 \\ 0 & -3 & 0 \\ 0 & 0 & 8 \end{bmatrix} \)
Only the diagonal positions \( (1,1), (2,2), (3,3) \) may have non-zero entries.
2.2. Diagonal Matrices Can Be Simple
- \( \begin{bmatrix} 7 \end{bmatrix} \) — a diagonal matrix of order \( 1 \times 1 \)
- \( \begin{bmatrix} 4 & 0 \\ 0 & 9 \end{bmatrix} \) — order \( 2 \times 2 \)
- \( \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \) — zero diagonal matrix
3. How to Identify a Diagonal Matrix
A matrix is diagonal if:
- It is a square matrix.
- Every entry that is not on the main diagonal is zero.
A quick test: scan the matrix — if all the non-diagonal elements are 0, it's diagonal.
4. Why Diagonal Matrices Are Important
Diagonal matrices are very easy to work with because their structure simplifies many operations:
- Adding or multiplying diagonal matrices is straightforward.
- Finding powers like \( D^2, D^3 \) is easy — just square or cube the diagonal entries.
- They are used in matrix decomposition and linear transformations.
- Many matrices can be converted into diagonal form using advanced methods (diagonalisation).
Because only the diagonal matters, computations become faster and cleaner.