Full rank matrix
A full-rank matrix refers to a matrix whose row rank is equal to the column rank. That is to say, there is no redundancy or linear correlation between the rows or columns in the matrix, and the rows and columns are linearly independent. An m × n matrix is defined if its row rank (the rank of the group of row vectors) is equal to its column rank (the rank of the group of column vectors), and the row rank (the rank of the column vector) is equal to the smaller value in the matrix (min(m, n) ), then this matrix is a full-rank matrix.
Give a simple example:
Consider a 3 × 3 matrix:
A = [ 1 2 3 4 5 6 7 8 9 ] A = \begin{bmatrix} 1 & 2 & 3 \ \ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} A=
147258369
We can use row transformations or look at the vectors in the matrix to determine their rank. This results in a matrix with a rank less than 3, when in fact the matrix has a rank of 2. Therefore, matrix A is not a full-rank matrix because its rank is smaller than the smallest dimension (3).
The important property of a full-rank matrix is that it can be viewed as a matrix without redundant information, which makes it very useful in many mathematical and engineering problems, such as solving systems of linear equations, matrix inversion, etc.
Positive definite matrix
A positive definite matrix refers to a matrix in which all eigenvalues in a symmetric matrix are positive. For an n × n real symmetric matrix A, if for all non-zero real vectors x, there is x T A x > 0 x^T A x > 0 xTAx>0, then the matrix A is said to be a positive definite matrix.
Simply put, for a positive definite matrix, when any non-zero real vector x is left multiplied twice by matrix A ( x T A x x^T A x xTAx), the result is always a positive number.
for example:
A = [ 2 − 1 − 1 4 ] A = \begin{bmatrix} 2 & -1 \\ -1 & 4 \end{bmatrix} A=[2−1−14]
For this 2 × 2 matrix A, it is a positive definite matrix. We can verify as follows:
- Calculate the eigenvalues of matrix A: det ( A − λ I ) = 0 \text{det}(A - \lambda I) = 0 det(A−λI)=0, the eigenvalue can be obtained by solving the characteristic equation: λ 1 = 1 \lambda_1 = 1 l1=1 sum λ 2 = 5 \lambda_2 = 5 l2=5, both eigenvalues are greater than zero.
- For any non-zero real vector x (for example, x = [ 1 1 ] x = \begin{bmatrix} 1 \\ 1 \end{bmatrix} x=[11]），计算 x T A x x^T A x xTAx，结果为正数： x T A x = [ 1 1 ] [ 2 − 1 − 1 4 ] [ 1 1 ] = 5 > 0 x^T A x = \begin{bmatrix} 1 & 1 \end{bmatrix} \begin{bmatrix} 2 & -1 \\ -1 & 4 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} = 5 > 0 xTAx=[11][2−1−14][11]=5>0。
Therefore, matrix A is a positive definite matrix, because its eigenvalues are all positive, and for all non-zero real vectors x, x T A x x^T A x < /span>xTAx始终大于零。