MATRIX OPERATIONS
The diagonal entries in an \(m\times n\) matrix \(A\) (\(a_{ij}\) are \(a_{11},a_{22},a_{33},\dots\) and they form the main diagonal of A. A diagonal matrix is a square n n matrix whose non-diagonal entries are zero. An example is the \(n\times n\) identity matrix \(I_n\). A \(m\times n\) matrix whose entries are all zero is a zero matrix and is written as \(0\). The size of a zero matrix is usually clear from the context.
Let \(A,B\), and \(C\) be matrices of the same size, and let rand \(s\) be scalars.
a. \(A+B=B+A\)
b. \(A+(B+C)=(A+B)+C\)
c. \(A+0=A\)
d. \((r+s)A=rA+sA\)
e. \(r(A+B)=rA+rB\)
f. \(r(sA)=(rs)A\)
The transpose of \(A\)(which is written as \(A^T\)) is defined as swapping columns and rows of \(A\).
The transpose of a product of matrices equals the product of their transposes in the reverse order : \((AB)^T=B^TA^T.\)
Inversity Matrices
MATRIXFACTORIZATIONS
At first,assume thatAis anm nmatrix that can be row reduced to echelon form,
without row interchanges.(Later,we will treat the general case.)ThenAcan be written
in the formADLU,whereLis anm mlower triangular matrix with 1’s on the
diagonal andU is anm nechelon form ofA.
Row reduce \(A\) into \(U\), we got \(E_p\dots E_2E_1A=U\). Then \(A=(E_1)^{-1}(E_2)^{-1}\dots (E_p)^{-1}U\). So, we got \(L=(E_1)^{-1}(E_2)^{-1}\dots (E_p)^{-1}.\)
标签：dots,matrix,written,anm,24.01,diagonal,002,zero,MATRICES From： https://www.cnblogs.com/zcr-blog/p/18604487