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Diagonal Matrix

A square matrix with nonzero entries only on the main diagonal -- the simplest matrix structure to invert, multiply, and reason about, and the target form of both spectral decomposition and a diagonal covariance matrix under independence.

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In Plain English

A diagonal matrix is about as simple as a matrix gets: every entry off the main diagonal is zero, so the matrix's only real content is the list of numbers running down its diagonal. That simplicity is exactly why statisticians care about them, a diagonal covariance matrix means the variables involved are all uncorrelated with each other, and many advanced techniques (like spectral decomposition) work by finding a way to rewrite a complicated matrix in terms of a simple diagonal one.

Definition

A diagonal matrix $D$ is a square matrix in which every off-diagonal entry is zero, $d_{ij} = 0$ for all $i \neq = j$ , so that $D = diag (d_{1}, d_{2}, \dots, d_{n})$ is fully specified by its $n$ diagonal entries. Diagonal matrices are trivial to multiply (entrywise on the diagonal), invert (the inverse is diagonal with entries $1/ d_{i}$ , provided every $d_{i} \neq = 0$ ), and raise to powers, and a diagonal covariance matrix corresponds exactly to a set of mutually uncorrelated random variables, since every off-diagonal covariance entry equals zero.

Formula

D = diag (d_{1}, d_{2}, \dots, d_{n})

Notation

d_{i}

—the i-th diagonal entry

diag (\cdot)

—notation for a diagonal matrix built from a list of entries

Properties

Diagonal matrices are the easiest matrices to compute with -- multiplying two diagonal matrices, inverting a diagonal matrix, or raising one to a power all reduce to doing the same simple operation to each diagonal entry independently, with no cross-terms to track.
A diagonal covariance matrix is the algebraic signature of mutual uncorrelatedness among a set of random variables -- every off-diagonal entry (each a pairwise covariance) is exactly zero, which is why 'independent' and 'diagonal covariance matrix' are used almost interchangeably in the multivariate normal case, where zero covariance and independence coincide.
Spectral decomposition rewrites any symmetric matrix as $QΛ Q^{⊤}$ , where $Λ$ is diagonal -- reducing a general matrix's behavior to a diagonal matrix's much simpler behavior is a recurring strategy throughout multivariate statistics, from PCA to the interpretation of a Fisher information matrix's eigenvalues.

At a Glance

Structuresquare, nonzero entries only on the main diagonal

Inverse (if all d_i != 0)diagonal, with entries 1/d_i

In a covariance matrixdiagonal means all variables are mutually uncorrelated

Target ofspectral/eigendecomposition, which rewrites a matrix around a diagonal core

Last updated September 8, 2026← Back to the Dictionary