Multiplying complex matrices and hermition

In summary, the original problem asks to solve for c in the equation S = Vc. The solution for c is either c = <V, S>/||V||^2 or c = (1/a)*<V, S>, where "a" is the value of the entries on the main diagonal of the Hermition matrix of V. However, there was some confusion as S should be a column matrix instead of a row matrix, which can be solved by transposing it.
  • #1
polaris90
45
0

Homework Statement



I have a 4x4 matrix V composed of complex numbers. I also have a 1x4 matrix S. The question asks to solve for c in S = Vc

Homework Equations



I learned that c = <V, S>/||V||^2, or c= (1/a)*<V, S> where "a" is the value of the entries on the main diagonal of the Hermition matrix of V.

The Attempt at a Solution



I don't know how to solve for c because I suppose that S should be a column matrix not a row matrix, unless I transpose it but that hasn't been defined. Some help would be appreciated.
 
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  • #2
polaris90 said:

Homework Statement



I have a 4x4 matrix V composed of complex numbers. I also have a 1x4 matrix S. The question asks to solve for c in S = Vc
I think there is a mistake in the problem statement. For Vc to be defined, c has to be 4 X something. If c is 4 X 1, then the product Vc will be 4 X 1.
polaris90 said:

Homework Equations



I learned that c = <V, S>/||V||^2, or c= (1/a)*<V, S> where "a" is the value of the entries on the main diagonal of the Hermition matrix of V.

The Attempt at a Solution



I don't know how to solve for c because I suppose that S should be a column matrix not a row matrix, unless I transpose it but that hasn't been defined. Some help would be appreciated.
 
  • #3
BTW, this should be posted in the Calculus & Beyond section. I am moving this thread to that section.
 
  • #4
Thanks for the reply, I was confirmed that the 1x4 should be transposed which solves the problem.
 

1. What are complex matrices?

Complex matrices are matrices with complex numbers as its entries. A complex number is a combination of a real number and an imaginary number, expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.

2. How do you multiply complex matrices?

To multiply complex matrices, you first need to make sure that the number of columns in the first matrix is equal to the number of rows in the second matrix. Then, use the standard matrix multiplication rule, treating the complex numbers as single entries. The result will be a new complex matrix.

3. What is the difference between a complex matrix and a hermitian matrix?

A complex matrix is a matrix with complex numbers as entries, while a Hermitian matrix is a special type of complex matrix where the entries satisfy the property that the conjugate of the entry in the i-th row and j-th column is equal to the entry in the j-th row and i-th column. In other words, the matrix is equal to its own conjugate transpose.

4. How do you compute the conjugate transpose of a complex matrix?

To compute the conjugate transpose of a complex matrix, you simply take the transpose of the matrix and then replace each entry with its complex conjugate. This is denoted by A* or A†.

5. Why is it important to understand multiplying complex matrices and Hermitian matrices?

Multiplying complex matrices and Hermitian matrices is fundamental in various fields of science, such as quantum mechanics and signal processing. These concepts are also used in many practical applications, such as computer graphics, image processing, and data compression. Understanding these concepts allows scientists to solve complex problems and develop advanced technologies.

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