matrix

Operations of Matrix

Transpose

$A^T$ and $(AB{)}^T=B^T{A}^T$

Conjugate transpose

$A^H$

Determinant

$$\det (\left[\begin{array}{cc}a& b\\ c& d\end{array}\right])=ad-cb$$ $$\det (\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right])=a(ei-fh)-b(di-fg)+c(dh-eg)$$

Properties of Matrix

Rank

• The rank of a matrix is equal to the number of non-zero rows if it is in Echelon Form.

Types of Matrix

• Singular, $\det (A)=0$. It’s inverse is not exists.
• Diagonal, $(\forall i,j)i\ne j⟹a_{ij}=0$.
• Tridiagonal, $(\forall i,j)|i-j|>1⟹a_{ij}=0$.
• Triangular, lower: $(\forall i,j)i>j⟹a_{ij}=0$; higher: $(\forall i,j)i<j⟹a_{ij}=0$.
• Symmetric, $A=A^T$.
• Skew-Symmetric, $A=-A^T$.
• Hermitian, $A=A^H$.
• Unitary, $A^{H}A=A{A}^H=I$.
• Normal $A=A^H$; $A^{H}A=A{A}^H$.
• Defective, Multiplicity $k>1$ with fewer than $k$ linearly independent corresponding Eigenvectors.
• Nondefective / diagonalizable, it has $X^{-1}AX=D$ where $X$ is Eigenvectors.

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• Orthogonal matrix, $A^{T}A=A{A}^T=I$.
• Symmetric positive definite matrix (SPD)
• Band matrix
• Vandermonde matrix
• Hessenberg matrix
• Jacobian matrix
• Hessian matrix