Need Help: New kind of Matrix formula?
In my book, I've:
Problem: Express the matrix [] as a sum of symmetric and skew symmetric matrices.
Okay, as far as I know, symmetric matrix remains the same even if we transpose it.
And the skew symmetric matrix is the same as it's -1 x the transpose of itself.
How do I suppose to solve this?
here [] =
| 2 -4 9 |
| 14 7 13 |
| 3 5 11 |
I think, first I should write the transpose of matrix which is the symmetric matrix and then write the transpose again multiplied by -1.
After that I can add both matrix and prove that it is equal to the [] matrix.
but in book, they used these formula,
B = 1/2 (A+A')
C = 1/2 (A-A')
I'm not sure what is this... and where it came from π¨
Anyone can explain me? My method is right or the method used in the book? β
Problem: Express the matrix [] as a sum of symmetric and skew symmetric matrices.
Okay, as far as I know, symmetric matrix remains the same even if we transpose it.
And the skew symmetric matrix is the same as it's -1 x the transpose of itself.
How do I suppose to solve this?
here [] =
| 2 -4 9 |
| 14 7 13 |
| 3 5 11 |
I think, first I should write the transpose of matrix which is the symmetric matrix and then write the transpose again multiplied by -1.
After that I can add both matrix and prove that it is equal to the [] matrix.
but in book, they used these formula,
B = 1/2 (A+A')
C = 1/2 (A-A')
I'm not sure what is this... and where it came from π¨
Anyone can explain me? My method is right or the method used in the book? β
Replies
-
ErAnushkaNo problem I got π
A direct proof is as follows:
Let X be the given square matrix
Let X=A+B where A is symmetric and B is skew symmetric
That is A'=A and B' = - B .......(1)
Therefore,
X'=A'+B' ( using the property of transpose of a matrix)
or, X'=A - B from (1) above
Now, as X is square , therefore, X+X' and X-X' are defined.
Therefore X+X'=A+B +A-B= 2A
and X-X'=A+B- (A-B) = 2B
Therefrore A=(X+X')/2
and B= (X-X')/2
It is easy to verify that A'=A and B'= - B
That is A is symmetric and B is skew symmetric
Therefore , X can be uniquely expressed as sum of a symmetric matrix and a skew symmetric matrix, which is
X =(X+X')/2 + (X-X')/2
I understood the green part π , but the red part is little confusingπ .. A'=A and B' = -B but how? we got only A=(X+X')/2 and B= (X-X')/2.... ??? π¨ -
silverscorpionI'm not sure what you are not getting here.. A = A' and B = -B' are assumptions that you made in the beginning, right? (ie., you have assumed that A is symmetric and B is skew symmetric)
You can probably take the transpose of (X+X')/2 and (X-X')/2, and check that they are indeed the same as A and -B.. but, here it's actually assumed that it is true. -
ErAnushka
Omg wait... I got it πsilverscorpionI'm not sure what you are not getting here.. A = A' and B = -B' are assumptions that you made in the beginning, right? (ie., you have assumed that A is symmetric and B is skew symmetric)
You can probably take the transpose of (X+X')/2 and (X-X')/2, and check that they are indeed the same as A and -B.. but, here it's actually assumed that it is true.
We are actually proving that any square matrix is a sum of symmetric matrix and skew symmetric matrix, so here, A is symmetric matrix and B is skew symmetric matrix.
We got both value, so,
A+B = (X+X')/2 + (X-X')/2
A+B = (X+X' + X-X') /2
A+B = 2X/2
A+B = X
Here X is a square matrix! OMG! I'm so stupid π
By the by thanks for the reply, I got the idea at the moment of reading this again π
You are reading an archived discussion.
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