A Clear Introduction to Diagonal Matrices

A diagonal matrix is a square matrix in which every entry away from the main (leading) diagonal is zero. The leading diagonal runs from the top-left corner of the matrix to the bottom-right corner, and these diagonal entries are the only positions that may contain non-zero values.

All off-diagonal entries must be zero. The diagonal entries themselves can be any real numbers, including zero. This strict structure is what makes diagonal matrices especially simple to analyse and compute with in linear algebra.


Examples of Diagonal Matrices

The general 2×2 diagonal matrix has the form:

(a   0)
(0   b)

The general 3×3 diagonal matrix has the form:

(a   0   0)
(0   b   0)
(0   0   c)

In both cases, the values on the leading diagonal (a, b, c, …) are the only entries that may be non-zero. Every position above or below this diagonal is fixed at 0.

The General n×n Diagonal Matrix

For an n×n diagonal matrix, the same pattern continues. We place values d₁, d₂, …, dā‚™ on the leading diagonal and set all other entries to zero. A convenient notation for this structure is: 

diag(d₁, d₂, …, dā‚™)

This expresses the idea of a matrix whose only potentially non-zero entries lie along the leading diagonal.

Why Diagonal Matrices Are Important

Diagonal matrices are fundamental in linear algebra because they make many operations easy. Multiplying a diagonal matrix by a vector simply scales each component of the vector by its corresponding diagonal entry. More broadly, when a matrix can be diagonalised, many problems—such as computing powers of a matrix—become dramatically simpler.

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