Orthogonal Matrices

$A\in \R_{n,n}$ is said orthogonal if the product between it and its transpose is the identity matrix:

$AA^T = I = A^TA$

An orthogonal matrix is always non-singular and its determinant is either 1 or -1

A matrix is orthogonal if and only if the ssum of the squares of the element of a row(column) is equal to 1 and the scalar product of any two arbitary rows(columns) is equal to 0: $\sum^n_{j=1}a^2_{i,j} =1$ $\forall i,j a_ia_j=0$

Rank of a Matrix

Rank - $A\in \R_{m,n}$ of matrix $A$, indicated as $\rho_A$ is the highest order of the non-singular submatrix $A_ρ \subset A$. So max value of rank in 3x3 matrices would be 3. If there is linear independence, then go to 2x2 and so forth. If $A$ is the null matrix then its rank is taken equal to 0

$A\in \R_{n,n}$ Matrix $A$ has ρ linearly independent rows(columns).

Rank is c

Sylvester's Lemma

$A\in \R_{n,n}$ and $B\in \R_{n,q}$. Let $A$ be non-singular and $ρ_B$ be the rank of the matrix $B$. Follows that the rank of the product matrix $AB$ is $ρ_B$

Law of Nullity

follows from the previous lemma $ρ_A$ and $ρ_B$ be the ranks and $ρ_{AB}$ be rank of the product AB. FOllows that: $ρ_{AB}\ge ρ_A+ρ_B-n$