11. Matrix spaces, rank 1 matrices and small world graphs
1. Matrix space
For the set \(\mathbb{R}^{n\times n}\) composed of real matrices of size \( n\times n\) , it satisfies the closure of addition and multiplication, so in this set Each element of can be analogized as a vector, this set is also a linear space, called matrix space.
\(\mathbb{R}^{n\times n}\) The most common subspaces are as follows:
(1) The set of all n-order real symmetric matrices \(S\) (symmetric matrix), \(\mathrm{dim}S=\frac 1 2 n(n+1)\)
(2) The set of all n-order real upper triangular matrices \(U\) (upper triangular matrix), \(\mathrm{dim}U=\frac 1 2 n(n-1)\)
(3) The set of all n-order real diagonal matrices \(D\) (diagonal matrix), \(\mathrm{dim}D=n\)
2. Intersection and sum of linear subspaces
For two linear subspaces \(W_1,W_2\) , their intersection and sum are defined as:
\[W_1\cap W_2=\{\alpha|\alpha\in W_1 \ \mathrm{ and } \ \alpha\in W_2\}\]
\[W_1+ W_2=\{\alpha+\beta|\alpha\in W_1 \ \mathrm{ and } \ \beta\in W_2\}\]
\[\mathrm{dim} W_1+\mathrm{dim} W_2-\mathrm{dim}(W_1 \cap W_2)=\mathrm{dim}(W_1+W_2)\]
For the \(S\) and \(U\) mentioned in the previous 1 , \(S\cap U=D\) , \(S+U=\mathbb R^{n\times n}\)
Note that the "union of linear subspaces ( \(W_1 \cup W_2\) )" is not necessarily a linear space, for example \(W_1\) represents a whole vector on a straight line passing through the origin, \(W_2\) represents a passing The overall vector on the plane of the origin, \(W_1 \cup W_2\) is the straight line inserted in this plane, then the sum of any non-zero vector in \(W_1\) and any non-zero vector in \(W_2\) The sum does not belong to \(W_1 \cup W_2\) , so the set is not closed for addition and is not a linear space.
3. Differential equations
For the differential equation \(\frac{d^2y}{dx^2}+y=0\) , its general solution is \(y=c_1sinx+c_2cosx\) , then the dimension of the solution space of this equation is 2, A set of basis of the solution space is \(\{sinx,cosx\}\) , the solution space of this equation is obviously closed to addition and multiplication, so it is also a linear space.
4. Some properties of matrices of rank 1
For the \(n\times m\) rank 1 matrix \(A\) , it has the following properties:
\(A=\alpha \beta^T\) (where \(\alpha\) is an n-dimensional column vector, \(\beta\) is an m-dimensional column vector)
In addition, any matrix with rank \(r\) can be expressed as the sum of \(r\) rank 1 homotype matrices.
5. Others
define a collection
\[S=\{v|v\in \mathbb{R}^4,\sum_{i=1}^4v_i=0\}\]
Obviously the set is closed to addition and multiplication and is a linear space.
This set can be regarded as the null space of the matrix \(A=(1,1,1,1)\) . \(r(A)=1\) , so \(N(A)=4-1=3\) . And \(\mathrm{dim}N(A^T)=0\) , \(N(A^T)=\{0\}\)