## Matrix: Types

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.

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# * Value of determinant of [A] and transpose of [A] is same i.e. lAl = lTrans. Al

Null Matrix:

Matrix having all the elements as zero.

This matrix transforms all vectors in the space to zero vector i.e. at the origin.

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Square Matrix:

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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.

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Diagonal Matrix:

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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.

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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:

â€‹â€‹Symmetric and Skew-Symmetric Matrix:

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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:

Orthogonal Matrix:

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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.

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* Value of determinant of an orthogonal matrix is equal to 1 or -1 i.e. lAl = 1 or -1.

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â€‹Idempotent Matrix:

For an idempotent matrix

Nilpotent Matrix:

â€‹For a nilpotent matrix

k is a positive integer.

Least positive integer for which above value is true is called index of the matrix.

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â€‹Involuntary Matrix:

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For an involuntary matrix

It means matrix is inverse of itself.

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â€‹â€‹Singular Matrix:

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A matrix is called a singular matrix if its determinant is zero i.e. lAl = 0.

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â€‹Triangular Matrix:

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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.

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* Determinant of triangular matrix is equal to product of diagonal elements.

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Complex Matrices:

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Matrix whose elements may be complex numbers.

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a) Conjugate of a Matrix:

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Matrix formed by conjugates of complex elements of original matrix.