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Transpose conjugate of a matrix and properties is considered one the most difficult concept.
11 Questions around this concept.
Let A and B be two symmetric matrices of order 3 .
Statement -1 : and are symmetric matrices.
Statement -2 : is symmetric matrix if matrix multiplication of A and B is commutative.
If a matrix A has a complex numbers as it’s elements, then the matrix obtained by replacing those complex number by their conjugates is called conjugate of the matrix A and it is denoted by . (If the element of a matrix is a + ib, then it is replaced by a - ib .)
Properties of the conjugate of a matrix:
If A and B are two matrices of the same order, then
i)
ii) where A and B are conformable for matrix addition.
iii) where A and B are conformable for multiplication.
iv) , where k is real or complex.
Transpose conjugate of a matrix and properties:
The transpose of a conjugate matrix A is called the transposed conjugate of A and is denoted by A?. The conjugate of the transpose of A is the same as the transpose of the conjugate of A
Properties of the transpose conjugate matrix:
If A and B are two matrices of the same order then
i) conjugate of a conjugate of matrix is the same as the original matrix itself,
In mathematical language (A?)? = A, which is quite obvious as we are reversing back the things which we did while taking conjugate at first time.
ii) (A + B)? = A? + B?, this is obvious if a matrix is conformable, as addition is done element-wise.
Iii )(kA)? = kA?, since multiplication in a matrix are elementwise, hence this is also obvious, as all elements multiplied before conjugate will be amplified in the way as after taking conjugate.
iv) (AB)? = B?A?, here A and B should be conformable for matrix multiplication.
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