Matrix ( 4x4) & Determinant
|Trace of a Matrix:Σi=1 to n aii Properties: tr (A+B) = tr(A) + tr(B)---(1) ; tr(cA) =c*tr(A) -----(2) where c is a scalar ; tr(AB) =tr(BA)......(3) tr(A) =tr(AT)--(4) which means that trace of a matrix and that of its transpose are identical.|
|Trace of a product of matrices: it behaves similarly to a dot
product of vectors.
Trace is invariant under cyclic permutations : tr(ABCD) = tr(BCDA)=tr(CDAB)=tr(DABC); This is known as cyclic property. But arbitrary permutations are not allowed.In general, tr(ABCD) != tr(ACBD); However, if 3 symmetric matrices are concerned, any permutation is allowed. For more than 3 factors, this is not true. Trace is similarity invariant, i.e. if P-1AP=A, tr(P-1AP)=tr(A).
If A is symmetric and B is anti-symmetric, tr(AB)=0;
|The trace corresponds to the derivative of the determinant. Suppose
where θ is the angle of rotation of the coordinate axis. These are one parameter family of linear transformations having determinant 1, so they preserve area. The derivative of this family at θ =0 , the identity rotation is anti-symmetric matrixa de
having trace zero. Trace is a linear operator and hence commutes with its derivative. d tr(A)=tr(dA);
The trace of a 2-by-2 complex matrix is used to classify Möbius transformations. First the matrix is normalized to make its determinant equal to one. Then, if the square of the trace is 4, the corresponding transformation is parabolic. If the square is in the interval [0,4), it is elliptic. Finally, if the square is greater than 4, the transformation is loxodromic. See classification of Möbius transformations.
|✍ The determinant of the adjoint of a square matrix, |adj. A| = |A|n-1
where n is the order of A.
✍ [adj.A]*[A] = |A| I which is a scalar matrix if |A| ≠ 0 and is a null matrix if |A| =0
✍ If A is a skew-symmetric matrix, |adj A| = 0
✍ If f(x) =[ cos x sinx 0 and f(y) = [ cos y siny 0 , then f(x)*f(y) = f(x+y) and [f(x)]-1 =f(-x) , [f(y)]-1 =f(-y)
-sinx cosx 0 -siny cosy 0 [f(x)]-1 * [f(y)]-1 =f(-x-y)
0 0 1 ] 0 0 1 ]
Another function say f(x) =ax and f(y)=ay , then f(x)*f(y) = f(x+y) and [f(x)]-1 =f(-x) , [f(y)]-1 =f(-y),[f(x)]-1 * [f(y)]-1 =f(-x-y)
hence there is some similarity between the above matrix and the exponential function.
|Matrix Norms : link
1-norm is the maximum of the absolute sum of respective columns, infinity-norm is the maximum of the absolute sum of the respective rows, euclidean- norm is the square root of the sum of the square of all the matrix elements.