Transpose of a Matrix:
Matrix formed by exchanging rows and corresponding columns of the original matrix. Transpose of a matrix A:
Hence; (a)ij of [A] = (a)ji of transpose of [A], where (a)ij is element in ith row and jth column and (a)ji is element in jth row and ith column.
* Value of determinant of [A] and transpose of [A] is same i.e. lAl = lTrans. Al
Matrix having all the elements as zero.
This matrix transforms all vectors in the space to zero vector i.e. at the origin.
Matrix having equal number of rows and columns. A 2 x 2 matrix is:
2 x 2 square matrix transforms a 2-D vector such that it remains in 2-D plane only. Likewise, a 3 x 3 square matrix transforms a 3-D vector such that remains in 3-D space only. Whereas a non-square matrix may transform a 2-D vector into 3-D vector and vice-versa.
A square matrix having only non-zero diagonal elements. All other elements of the matrix are zero.
A diagonal matrix is called 'Scalar Matrix' when a = b. A scalar matrix only stretches or shrinks a vector along its span.
A diagonal matrix is called a 'Unit/Identity Matrix' if a = b = 1. Unit matrix is represented by I and does nothing to the vector after transformation. A 3x3 unit matrix is:
Symmetric and Skew-Symmetric Matrix:
A square matrix is called 'Symmetric' if
(a)ij = (a)ji
Hence; Trans.[A] = [A]
A 3 x 3 symmetric matrix is:
For a 'Skew-Symmetric Matrix'
(a)ij = -(a)ji and (a)ii = 0
Hence;Trans.[A] = -[A]
A 3 x 3 skew-symmetric matrix is:
A square matrix satisfying the condition
An orthogonal matrix transforms a space such that axes always remains perpendicular to each other and magnitude of unit vectors remains unchanged (rotation/reflection). Consider an orthogonal matrix:
For the above matrix
so i(transformed) and j(transformed) are perpendicular to each other and their magnitude is equal to 1.
* Value of determinant of an orthogonal matrix is equal to 1 or -1 i.e. lAl = 1 or -1.
For an idempotent matrix
For a nilpotent matrix
k is a positive integer.
Least positive integer for which above value is true is called index of the matrix.
For an involuntary matrix
It means matrix is inverse of itself.
A matrix is called a singular matrix if its determinant is zero i.e. lAl = 0.
A square matrix having all elements below the main diagonal as zero is called 'Upper Triangular (UT) Matrix'. For upper triangular matrix (a)ij = 0 for all i>j.
A square matrix having all elements above the main diagonal as zero is called 'Lower Triangular (LT) Matrix'. For lower triangular matrix (a)ij = 0 for all i<j.
* Transpose of a LT Matrix is a UT matrix.
* Determinant of triangular matrix is equal to product of diagonal elements.
Matrix whose elements may be complex numbers.
a) Conjugate of a Matrix:
Matrix formed by conjugates of complex elements of original matrix.