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 higherdimensional parallelepiped.
Geometric Interpretation of Determinant
Geometric Interpretation of Determinant
Geometric Interpretation of Determinant
The determinant of a 3×3 matrix
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 ).
Three Vectors
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.
Geometric Interpretation of 3x3 Determinant
3-by-3 matrices
The volume of this Parallelepiped is the absolute value of the
determinant of the matrix formed by the rows r1, r2, and r3.
The determinant of a 3×3 matrix
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([A B]) = 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))
Inverse
An n-by-n (square) matrix A is called invertible if there exists an nby-n matrix B such that
where In denotes the n-by-n identity matrix and the multiplication
used is ordinary matrix multiplication. If this is the case, then the
matrix B is uniquely determined by A and is called the inverse of
A, denoted by A−1. It follows from the theory of matrices that if
For finite square matrices A and B, then also
Right and Left Inverse
Right and Left Inverse
Non-square matrices (m-by-n matrices for which m ≠ n) do
not have an inverse. However, in some cases such a matrix
may have a left inverse or right inverse. If A is m-by-n and
the rank of A is equal to n, then A has a left inverse: an n-bym matrix B such that BA = I. If A has rank m, then it has a
right inverse: an n-by-m matrix B such that AB = I.
A square matrix that is not invertible is called singular or
degenerate. A square matrix is singular if and only if its
determinant is 0. Singular matrices are rare in the sense that
if you pick a random square matrix, it will almost surely not
be singular.
Properties of Inverse Matrix
Properties of Inverse Matrix
A is invertible.
A is row-equivalent to the n-by-n identity matrix In.
A is column-equivalent to the n-by-n identity matrix In.
det A ≠ 0.
rank A = n.
The equation Ax = 0 has only the trivial solution x = 0 (i.e., Null
A = {0})
The equation Ax = b has exactly one solution for each b in Kn,
(x ≠ 0).
The columns of A are linearly independent.
There is an n by n matrix B such that AB = In = BA.
The transpose AT is an invertible matrix (hence rows of A are
linearly independent, The number 0 is not an eigenvalue of A.
The matrix A can be expressed as a finite product of elementary
matrices.
Properties of Inverse Matrix
Inverse of a Matrix
Inverse of a Matrix
Example
Partitioned Matrices
Block Matrices
Properties of Block Matrices
Properties of Block Matrices
Sum and Difference of Partitioned Matrices
Product of Partitioned Matrices
Inverse of Partitioned Matrices
Inverse of Partitioned Matrices
Inverse by Partitioning
Inverse by Partitioning
Inverse by Partitioning
Properties
Properties
Properties
Properties
Properties
Properties
Properties
Properties
Properties