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Diagonal of a Matrix / Trace / Triangular Matrix

Table of contents
  1. Principal diagonal
  2. Trace
    1. Invariant under Cyclic Permutation
  3. Diagonal Matrix
    1. Properties of a Diagonal Matrix
  4. Triangular Matrix
    1. Upper Triangular Matrix
    2. Lower Triangular Matrix
  5. Rectangular Diagonal 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 $\boldsymbol{A}$ is the sum of its principal diagonal elements:

$$ \tr(\boldsymbol{A}) = \sum_{i=1}^n a_{ii} $$

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

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

    This property also holds for non-square matrices.

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

Invariant under Cyclic Permutation

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

$$ \tr(\boldsymbol{ABC}) = \tr(\boldsymbol{BCA}) = \tr(\boldsymbol{CAB}) $$


Diagonal Matrix

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

\[A = \begin{bmatrix} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{nn} \end{bmatrix}\]

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 $B = diag(b_{11}\cdots, b_{nn})$:

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

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:

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

Example:

\[\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}\]

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:

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

Example:

\[\begin{bmatrix} 1 & 0 & 0 \\ 2 & 3 & 0 \\ 4 & 5 & 6 \end{bmatrix}\]

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

If $m > n$:

\[\boldsymbol{A} = \begin{bmatrix} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{nn} \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 \end{bmatrix}\]

If $m < n$:

\[\boldsymbol{A} = \begin{bmatrix} a_{11} & 0 & \cdots & 0 & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{mm} & 0 & \cdots & 0 \end{bmatrix}\]

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

$a_{ii}$ can be zero.