Diagonal of a Matrix / Trace / Triangular Matrix

Principal diagonal

The principal diagonal of a square matrix is the diagonal from the upper left to the lower right.

$$\{a_{ij}\mid i=j\}$$

Also called the main diagonal.

Each element $a_{ii}$ is called a principal diagonal element.

Trace

A trace of a square matrix $\mathit{A}$ is the sum of its principal diagonal elements:

$$\mathrm{tr}(\mathit{A})=\sum _{i=1}^n{a}_{ii}$$

For square matrices $\mathit{A}$ and $\mathit{B}$:

• $\mathrm{tr}(\mathit{A}+\mathit{B})=\mathrm{tr}(\mathit{A})+\mathrm{tr}(\mathit{B})$

• $\mathrm{tr}(\mathit{A}\mathit{B})=\mathrm{tr}(\mathit{B}\mathit{A})$

  This property also holds for non-square matrices.

• $\mathrm{tr}(\lambda \mathit{A})=\lambda \mathrm{tr}(\mathit{A})$
• $\mathrm{tr}({\mathit{I}}_n)=n$

Invariant under Cyclic Permutation

Coming from the above properties, for matrices $\mathit{A},\mathit{B},\mathit{C}$, whose product is defined but are not necessarily square matrices,

$$\mathrm{tr}(\mathit{A}\mathit{B}\mathit{C})=\mathrm{tr}(\mathit{B}\mathit{C}\mathit{A})=\mathrm{tr}(\mathit{C}\mathit{A}\mathit{B})$$

Diagonal Matrix

A diagonal matrix is a square matrix whose non-principal diagonal elements are all zero.

Often denoted by:

$$A=diag(a_{11},a_{22},\cdots ,a_{nn})$$

Properties of a Diagonal Matrix

For diagonal matrices $A=diag(a_{11}\cdots ,a_{nn})$ and $B=diag(b_{11}\cdots ,b_{nn})$:

• $A+B=diag(a_{11}+b_{11},\cdots ,a_{nn}+b_{nn})$
• $AB=diag(a_{11}{b}_{11},\cdots ,a_{nn}{b}_{nn})$

Triangular Matrix

For both upper and lower triangular matrices, if the principal diagonal elements are all 1, then the matrix is called a unit (upper/lower) triangular matrix.

Upper Triangular Matrix

An upper triangular matrix is a square matrix whose non-principal diagonal elements below the principal diagonal are all zero.

Upper triangular matrix, often denoted by $U_n$ is:

$$U_n=[u_{ij}{]}_{n\times n}\text{where}\mathrm{\forall }i>j,u_{ij}=0$$

Example:

Lower Triangular Matrix

An lower triangular matrix is a square matrix whose non-principal diagonal elements above the principal diagonal are all zero.

Lower triangular matrix, often denoted by $L_n$ is:

$$L_n=[l_{ij}{]}_{n\times n}\text{where}\mathrm{\forall }i<j,l_{ij}=0$$

Example:

Rectangular Diagonal Matrix

Sometimes you might see people say rectangular diagonal matrix which refers to a non-square matrix that constains a diagonal matrix as a submatrix, and the rest of the elements are all zero.

Suppose $\mathit{A}\in {\mathbb{R}}^{m\times n}$.

If $m>n$:

If $m<n$:

You’ll see this in the context of singular value decomposition.

$a_{ii}$ can be zero.