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Inverse and Partition of Matrices and their Applications in Statistics. Professor Dr. S. K. Bhattacharjee Department of Statistics University of Rajshahi, Bangladesh. Geometric Interpretation of Determinant.
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Inverse and Partition of Matrices and their Applications in Statistics Professor Dr. S. K. Bhattacharjee Department of Statistics University of Rajshahi, Bangladesh
Geometric Interpretation of Determinant The determinant has an important geometric interpretation as the area of a parallelogram, and more generally as the volume of a higher-dimensional parallelepiped.
Properties Of determinant • If one of the two vectors is a scalar multiple of the other, this determinant is nil. • if one multiplies one of the two vectors by a scalar , the whole determinant is multiplied by that same scalar (since the corresponding area is multiplied by the scalar (in absolute value). • If a vector z is added to u, simply add the
Dot Product • The dot product of two vectors a = [a1, a2, ... , an] and b = [b1, b2, ... , bn] is defined as:
Cross Product • The cross product, also called the vector product, is an operation on two vectors. • The cross product of two vectors produces a third vector which is perpendicular to the plane in which the first two lie. • The cross product, A x B, gives a third vector, say C, whose tail is also at the same point as those of A and B.
Cross Product • . The vector C points in a direction perpendicular (or normal) to both A and B. The direction of C depends on the Right Hand Rule.
Cross Product • the cross product of A and B can be expressed as • A x B = A B sin(θ) • The cross product requires both of the vectors to be three dimensional vectors. • The result of a dot product is a number and the result of a cross product is a vector!
Cross Product Let two vectors a=(a1 a2 a3 ) and b= (b1 b2 b3). axb=(a2b3 –a3b2, a3b1 –a1b3 ,a1b2 –a2b1 ).
Geometric Interpretation of Determinant • The absolute value of the determinant together with the sign becomes the oriented area of the parallelogram. The oriented area is the same as the usual area, except that it is negative when the angle from the first to the second vector defining the parallelogram turns in a clockwise direction.
Properties of Determinant • det(AT) = det(A) • det(cA) = cn det(A) • det(Ak) = (det(A))k , • det(A[nxn])=0 iff rank(Anxn)<n.
Properties of Determinant • Interchanging any pair of columns of a matrix multiplies its determinant by -1(likewise rows). • Multiplying any column of a matrix by c multiplies its determinant by c (likewise rows). • Adding any multiple of one column onto another column leaves the determinant unaltered (likewise rows). • det(A) = 0 iff the columns of A are linearly dependent (likewise rows).
Properties of Determinant • When we interchange two rows of a matrix, the sign of the determinant changes, • When we add a scalar multiple of 1 row to another row of the matrix, the determinant stays the same, and • When we multiply a row by a non-zero scalar, the determinant is multiplied by the same scalar.
Properties of Determinant det(A) = 0 if two columns are identical (likewise rows). det(A) = 0 if any column consists entirely of zeros (likewise rows). The determinant of a diagonal or triangular matrix is the product of its diagonal elements. The determinant of a unitary matrix has an absolute value of 1. The determinant of an orthogonal matrix is ±1. det(AB) = det(A) det(B)
Properties of Rank • rank(X-) >= rank(X). • rank(X)=rank(X-) iff X is also the generalized inverse of X- ( i.e. X-XX-=X-.). • XX- and X-X are idempotent and have the same rank as X. • rank(A[mxn]) <= min(m,n). • rank(A[mxn]) = n iff its columns are linearly independent. • rank(A) = rank(AT) • rank(A) = maximum number of linearly independent columns (or rows) of A.
Properties of Rank • det(A[nxn])=0 iff rank(A[nxn])<n. • rank(A + B) <= rank(A) + rank(B) • rank([AB]) = rank(A) + rank(B – AA-B) where A- is a generalized inverse of A. rank([A; C]) = rank(A) + rank(C – CA-A) • rank(AB) + rank(BC) <= rank(B) + rank(ABC) rank(A[mxn]) + rank(B) - n <= rank(AB) <= min(rank(A), rank(B))
