Assignment 02 pr

Matrix at a glance

Matrix Type	Condition (LaTeX)	Description
Symmetric	$A = A^T$	Equal to its transpose.
Skew-Symmetric	$A = -A^T$	Equal to negative transpose.
Orthogonal	$A^T = A^{-1}$	Transpose equals inverse.
Idempotent	$A^2 = A$	Square equals itself.
Involutory	$A^2 = I$	Square equals Identity.
Nilpotent	$A^k = 0$	Power $k$ results in Zero matrix.
Singular	$	A
Upper Triangular	$a\_{ij} = 0 \text{ for all } i > j$	All elements below the main diagonal are zero.
Lower Triangular	$a\_{ij} = 0 \text{ for all } i < j$	All elements above the main diagonal are zero.
Hermitian	$A = A^\dagger$	Equal to conjugate transpose.
Skew-Hermitian	$A = -A^\dagger$	Equal to negative conjugate transpose.
Unitary	$AA^\dagger = I$	Conjugate transpose is inverse.

Sum of a symmetric or skew symmetric

Orthogonal Matrix

Cramer's Rules

Matrix Type	Condition (LaTeX)	Description
Symmetric	\(A = A^T\)	Equal to its transpose.
Skew-Symmetric	\(A = -A^T\)	Equal to negative transpose.
Orthogonal	\(A^T = A^{-1}\)	Transpose equals inverse.
Idempotent	\(A^2 = A\)	Square equals itself.
Involutory	\(A^2 = I\)	Square equals Identity.
Nilpotent	\(A^k = 0\)	Power \(k\) results in Zero matrix.
Singular	$	A
Upper Triangular	\(a\_{ij} = 0 \text{ for all } i > j\)	All elements below the main diagonal are zero.
Lower Triangular	\(a\_{ij} = 0 \text{ for all } i < j\)	All elements above the main diagonal are zero.
Hermitian	\(A = A^\dagger\)	Equal to conjugate transpose.
Skew-Hermitian	\(A = -A^\dagger\)	Equal to negative conjugate transpose.
Unitary	\(AA^\dagger = I\)	Conjugate transpose is inverse.