Every square matrix is uniquely expressible asa) The sum of a Hermitian Matrix and a Skew Hemitian matrixb) A + iB, where A and B both are symmetricc) A + iB, where A and B are real Skew symmetric matricesd) The sum of a symmetric and a Hermitian MatricesCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

A

Answer

Explanation:

To understand why option A is the correct answer, let's first define what a Hermitian matrix and a skew Hermitian matrix are:

Hermitian Matrix:
A Hermitian matrix is a square matrix that is equal to its conjugate transpose. In other words, if A is a Hermitian matrix, then A = A*.

Skew Hermitian Matrix:
A skew Hermitian matrix is a square matrix that is equal to the negative of its conjugate transpose. In other words, if B is a skew Hermitian matrix, then B = -B*.

Now, let's consider a general square matrix C. We can express C as the sum of a Hermitian matrix and a skew Hermitian matrix as follows:

C = (C + C*)/2 + (C - C*)/2i

Let's break it down step by step:

Step 1: C + C*
Taking the sum of a matrix and its conjugate transpose gives us a Hermitian matrix. This is because the diagonal elements remain the same, and the off-diagonal elements get added with their complex conjugates, resulting in real numbers.
Let's call this Hermitian matrix A.

Step 2: C - C*
Taking the difference of a matrix and its conjugate transpose gives us a skew Hermitian matrix. This is because the diagonal elements remain the same, and the off-diagonal elements get subtracted with their complex conjugates, resulting in imaginary numbers.
Let's call this skew Hermitian matrix B.

Step 3: Dividing by 2 and i
Dividing both A and B by 2 gives us their average, which is a symmetric matrix. Dividing B by i gives us another matrix, which is also symmetric.
Let's call the symmetric matrix (A + B)/2 as D and the symmetric matrix B/i as E.

So, we have:
C = (C + C*)/2 + (C - C*)/2i
= A/2 + B/2i
= D + E

Hence, any square matrix C can be uniquely expressed as the sum of a Hermitian matrix D and a skew Hermitian matrix E. Therefore, option A is the correct answer.

Similar Mathematics Doubts

Mark the incorrect statement

[A] is a square matrix which is neither symmetric nor skew-symmetric and [A]T is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]T and [D] = [A] -[A]T, respectively. Which of the following statements is True?

A square matrix A such that AT = -A, is called a

If A and B are two odd order Skew — symmetric matrices such that AB = BA, then what is the matrix AB?

Which of the following(s) is/are correct?

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