Square Matrices Types

	Identity	Zero	Scalar	Diagonal	Symmetric	Skew-symmetric	Upper Triangular	Lower Triangular
Identity Diagonal matrix with all 1's on main diagonal, 0's elsewhere	Yes	Never	Yes	Yes	Yes	Never	Yes	Yes
Zero All elements are 0	Never	Yes	Yes	Yes	Yes	Yes	Yes	Yes
Scalar Diagonal matrix with same value on main diagonal	May Be	May Be	Yes	Yes	Yes	May Be	Yes	Yes
Diagonal Only non-zero elements on main diagonal	May Be	May Be	May Be	Yes	Yes	May Be	Yes	Yes
Symmetric Equal to its own transpose (aᵢⱼ = aⱼᵢ)	May Be	May Be	May Be	May Be	Yes	May Be	May Be	May Be
Skew-symmetric Negative of its own transpose (aᵢⱼ = -aⱼᵢ)	Never	May Be	May Be	May Be	May Be	Yes	May Be	May Be
Upper Triangular All elements below main diagonal are 0	May Be	May Be	May Be	May Be	May Be	May Be	Yes	May Be
Lower Triangular All elements above main diagonal are 0	May Be	May Be	May Be	May Be	May Be	May Be	May Be	Yes

This table shows reciprocal
relations between different types of
square matrices.
Click on Types of matricfes
to see detailed explanations

Identity Matrix

Matrix where

a_{ij} = 1

i = j

and

a_{ij} = 0

i \neq j

\begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix}

✅ Always Identity: The identity matrix is itself by definition - this is not a relationship but a self-evident fact since $I$ consists of 1s on diagonal and 0s elsewhere
❌ Can never be Zero: The identity matrix must have 1s on its diagonal, while the zero matrix must have all elements as 0, making it impossible for an identity matrix to be a zero matrix
✅ Is a Scalar Matrix: Since a scalar matrix has form $kI$ where $k$ is a constant, identity matrix is simply a scalar matrix where $k=1$
✅ Always Diagonal: By definition, identity matrix has all off-diagonal elements as 0, which is exactly what makes a matrix diagonal
✅ Is Symmetric: An identity matrix satisfies $I^T = I$ since reflecting across diagonal doesn't change 1s and 0s positions
❌ Cannot be Skew-symmetric: A skew-symmetric matrix requires $a_{ij} = -a_{ji}$ and diagonal elements must be 0, but identity has 1s on diagonal
✅ Is Upper/Lower Triangular: Since all elements below AND above main diagonal are 0, identity matrix satisfies both upper and lower triangular definitions simultaneously

Zero Matrix

Matrix where

a_{ij} = 0

for all

i,j

\begin{pmatrix} 0 & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 \end{pmatrix}

❌ Never Identity: A zero matrix has all elements as 0, while identity matrix requires 1s on the diagonal
✅ Always Zero: By definition, all elements are 0
✅ Is Scalar: Zero matrix is a special case of scalar matrix where $k=0$
✅ Is Diagonal: All off-diagonal elements are 0, satisfying diagonal matrix definition
✅ Is Symmetric: Reflecting zero matrix across diagonal still gives all zeros, so $O^T = O$
✅ Is Skew-symmetric: Zero matrix satisfies $a_{ij} = -a_{ji}$ since $0 = -0$
✅ Is Upper/Lower Triangular: All elements above and below diagonal are 0, satisfying both triangular definitions

Scalar Matrix

Matrix where

a_{ij} = k

i = j

and

a_{ij} = 0

i \neq j

\begin{pmatrix} k & 0 & \cdots & 0 \\ 0 & k & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & k \end{pmatrix}

❔ May Be Identity: A scalar matrix becomes identity when $k=1$
❔ May Be Zero: A scalar matrix becomes zero matrix when $k=0$
✅ Always Scalar: By definition, it has same value $k$ on main diagonal and 0s elsewhere
✅ Always Diagonal: All off-diagonal elements are 0, making it diagonal by definition
✅ Is Symmetric: Since $a_{ij} = a_{ji} = 0$ for all $i \neq j$ , and diagonal elements are equal
❔ May Be Skew-symmetric: Only becomes skew-symmetric when $k=0$ (zero matrix case)
✅ Is Upper/Lower Triangular: All elements below and above diagonal are 0, satisfying both triangular definitions

Diagonal Matrix

Matrix where

a_{ij} = 0

for all

i \neq j

\begin{pmatrix} d_1 & 0 & \cdots & 0 \\ 0 & d_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & d_n \end{pmatrix}

❔ May Be Identity: Becomes identity matrix when all diagonal entries are 1
❔ May Be Zero: Becomes zero matrix when all diagonal entries are 0
❔ May Be Scalar: Becomes scalar matrix when all diagonal entries are equal
✅ Always Diagonal: By definition, all off-diagonal elements are 0
✅ Is Symmetric: Since $a_{ij} = a_{ji} = 0$ for all $i \neq j$
❔ May Be Skew-symmetric: Only when all diagonal entries are 0 (zero matrix case)
✅ Is Upper/Lower Triangular: All elements below and above diagonal are 0
Read more about diagonal matrices.

Symmetric Matrix

Matrix where

a_{ij} = a_{ji}

for all

i,j

\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{12} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{1n} & a_{2n} & \cdots & a_{nn} \end{pmatrix}

❔ May Be Identity: Only when it's a 1-diagonal matrix
❔ May Be Zero: Only when all elements are 0
❔ May Be Scalar: Only when all off-diagonal elements are 0 and diagonal elements are equal
❔ May Be Diagonal: Only when all off-diagonal elements are 0
✅ Always Symmetric: By definition, $a_{ij} = a_{ji}$ for all $i,j$
❔ May Be Skew-symmetric: Only when all elements are 0 (zero matrix case)
❔ May Be Upper/Lower Triangular: Only when all off-diagonal elements are 0

Skew-Symmetric Matrix

Matrix where

a_{ij} = -a_{ji}

for all

i,j

\begin{pmatrix} 0 & a_{12} & \cdots & a_{1n} \\ -a_{12} & 0 & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ -a_{1n} & -a_{2n} & \cdots & 0 \end{pmatrix}

❌ Never Identity: Cannot have 1s on diagonal since diagonal elements must be 0
❔ May Be Zero: When all elements are 0
❔ May Be Scalar: Only when $k=0$ (zero matrix case)
❔ May Be Diagonal: Only when all elements are 0
❔ May Be Symmetric: Only when all elements are 0
✅ Always Skew-symmetric: By definition, $a_{ij} = -a_{ji}$ and diagonal elements = 0
❔ May Be Upper/Lower Triangular: Only when all elements are 0

Upper Triangular Matrix

Matrix where

a_{ij} = 0

for all

i > j

\begin{pmatrix} u_{11} & u_{12} & \cdots & u_{1n} \\ 0 & u_{22} & \cdots & u_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & u_{nn} \end{pmatrix}

❔ May Be Identity: Only when diagonal elements are 1 and upper elements are 0
❔ May Be Zero: Only when all elements are 0
❔ May Be Scalar: Only when upper elements are 0 and diagonal elements are equal
❔ May Be Diagonal: Only when all upper elements are 0
❔ May Be Symmetric: Only when all upper elements match corresponding lower elements (which are 0)
❔ May Be Skew-symmetric: Only when all elements are 0
✅ Always Upper Triangular: By definition, all elements below diagonal are 0
❌ Never Lower Triangular: Unless all non-diagonal elements are 0 (diagonal matrix case)

Lower Triangular Matrix

Matrix where

a_{ij} = 0

for all

i < j

\begin{pmatrix} l_{11} & 0 & \cdots & 0 \\ l_{21} & l_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ l_{n1} & l_{n2} & \cdots & l_{nn} \end{pmatrix}

❔ May Be Identity: Only when diagonal elements are 1 and lower elements are 0
❔ May Be Zero: Only when all elements are 0
❔ May Be Scalar: Only when lower elements are 0 and diagonal elements are equal
❔ May Be Diagonal: Only when all lower elements are 0
❔ May Be Symmetric: Only when all lower elements match corresponding upper elements (which are 0)
❔ May Be Skew-symmetric: Only when all elements are 0
❌ Never Upper Triangular: Unless all non-diagonal elements are 0 (diagonal matrix case)
✅ Always Lower Triangular: By definition, all elements above diagonal are 0

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Square Matrices Types

Identity Matrix

Zero Matrix

Scalar Matrix

Diagonal Matrix

Symmetric Matrix

Skew-Symmetric Matrix

Upper Triangular Matrix

Lower Triangular Matrix