Skip to main content

Matrices

Properties​

Prove: (AB)′=(B′A′)\bold{(AB)'}=\bold{(B'A')}

Proof:

Prove: (A−1)′=(A′)−1\bold{(A^{-1})'}=\bold{(A')^{-1}}

Proof:

(A−1)′A′=(AA−1)′=I′=I  ⟹  (A−1)′=(A′)−1■\begin{align*} \bold{(A^{-1})'A'}&=\bold{(AA^{-1})'}\\ &=\bold{I'}\\ &=\bold{I}\\ \implies \bold{(A^{-1})'}&=\bold{(A')^{-1}}\hspace{20px}\blacksquare\\ \end{align*}

Prove: (AB)−1=B−1A−1\bold{(AB)^{-1}}=\bold{B^{-1}A^{-1}}.

Proof:

(AB)(AB)−1=IA−1(AB)(AB)−1=A−1I(IB)(AB)−1=A−1IB(AB)−1=A−1B−1B(AB)−1=B−1A−1(AB)−1=B−1A−1■\begin{align*} \bold{(AB)(AB)^{-1}}&=\bold{I}\\ \bold{A^{-1}(AB)(AB)^{-1}}&=\bold{A^{-1}I}\\ \bold{(IB)(AB)^{-1}}&=\bold{A^{-1}I}\\ \bold{B(AB)^{-1}}&=\bold{A^{-1}}\\ \bold{B^{-1}B(AB)^{-1}}&=\bold{B^{-1}A^{-1}}\\ \bold{(AB)^{-1}}&=\bold{B^{-1}A^{-1}}\hspace{20px}\blacksquare\\ \end{align*}

Note: Existence of (AB)−1\bold{(AB)^{-1}} does not imply that B−1\bold{B^{-1}} and A−1\bold{A^{-1}} exist.

Prove: E[Tr(X)]=Tr(E[X])\mathbb{E}[\text{Tr}(\bold{X})]=\text{Tr}(\mathbb{E}[\bold{X}]).

Tr(X)=∑i=1nxii,E[Tr(X)]=E[∑i=1nxii],E[Tr(X)]=∑i=1nE[xii].\begin{align*} \text{Tr}(\bold{X})&=\sum_{i=1}^nx_{ii},\\ \mathbb{E}[\text{Tr}(\bold{X})]&=\mathbb{E}[\sum_{i=1}^nx_{ii}],\\ \mathbb{E}[\text{Tr}(\bold{X})]&=\sum_{i=1}^n\mathbb{E}[x_{ii}].\\ \end{align*}

∑i=1nE[xii]\sum_{i=1}^n\mathbb{E}[x_{ii}] is equivalent to Tr(E[X])\text{Tr}(\mathbb{E}[\bold{X}]), therefore

E[Tr(X)]=∑i=1nE[xii]=Tr(E[X]).■\begin{align*} \mathbb{E}[\text{Tr}(\bold{X})]&=\sum_{i=1}^n\mathbb{E}[x_{ii}]=\text{Tr}(\mathbb{E}[\bold{X}]).\hspace{15px}\blacksquare\\ \end{align*}

Rank of a Matrix​

Column Space: The column space of a matrix is the vector space that is spanned by its column vectors.

For example

A=[156268718]\bold{A}= \begin{bmatrix} 1 & 5 & 6\\ 2 & 6 & 8\\ 7 & 1 & 8 \end{bmatrix}

here the third column is the sum of the first two hence the column space of this matrix is a two-dimensional subspace of R3\mathbb{R}^3.

Column Rank: The column rank of a matrix is the dimension of the vector space that is spanned by its column vectors. The column rank of the following matrix is 22.

A=[156268718]\bold{A}= \begin{bmatrix} 1 & 5 & 6\\ 2 & 6 & 8\\ 7 & 1 & 8 \end{bmatrix}

Row Rank: The row rank of a matrix is the dimension of the vector space that is spanned by its row vectors. The row rank of the following matrix is 33.

B=[123515645314]\bold{B}= \begin{bmatrix} 1 & 2 & 3\\ 5 & 1 & 5\\ 6 & 4 & 5\\ 3 & 1 & 4 \end{bmatrix}

The notation of row rank of matrix B\bold{B} is: rank B′\text{rank }\bold{B'}

Theorem: The column rank and row rank of a matrix are equal.

Column rank = Row rank = Rank of the matrix. That is

rank A=rank A′\text{rank }\bold{A}=\text{rank }\bold{A'}

Full column rank: If the column rank of a matrix happens to equal the number of columns it contains, then the matrix is said to have full column rank.

Full row rank: If the row rank of a matrix happens to equal the number of rows it contains, then the matrix is said to have full row rank.

Property:

rank A≤min⁡(number of rows, number of columns)\text{rank }\bold{A}\leq\min(\text{number of rows, number of columns})