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Sagot :
Certainly! Let's carefully solve for [tex]\( X \)[/tex] in the given matrix equation:
[tex]\[ \left(B\left(X^t - I\right)^t A\right)^t + B = A^t \][/tex]
We will solve this step-by-step.
### Step 1: Transpose the entire equation
First, we need to eliminate the outer transpose by transposing the entire equation:
[tex]\[ \left( \left(B\left(X^t - I\right)^t A\right)^t \right)^t + B^t = \left(A^t\right)^t \][/tex]
Since transposing twice returns the original matrix, we have:
[tex]\[ B\left(X^t - I\right)^t A + B^t = A \][/tex]
### Step 2: Rearrange the term involving [tex]\( B \)[/tex]
We need to simplify the equation:
[tex]\[ B\left(X^t - I\right)^t A + B = A^t \][/tex]
### Step 3: Subtract [tex]\( B \)[/tex] from both sides
Move [tex]\( B \)[/tex] to the right-hand side of the equation:
[tex]\[ B\left(X^t - I\right)^t A = A^t - B \][/tex]
### Step 4: Isolate the term involving [tex]\( X \)[/tex]
Next, we need to deal with the term [tex]\(\left(X^t - I\right)^t\)[/tex]. As [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are non-singular, they have inverses [tex]\( A^{-1} \)[/tex] and [tex]\( B^{-1} \)[/tex]. We will use these inverses to isolate the term [tex]\(\left(X^t - I\right)^t\)[/tex]:
Multiply both sides by [tex]\( B^{-1} \)[/tex] from the left:
[tex]\[ \left(X^t - I\right)^t A = B^{-1}(A^t - B) \][/tex]
Next, multiply both sides by [tex]\( A^{-1} \)[/tex] from the right:
[tex]\[ \left(X^t - I\right)^t = B^{-1}(A^t - B) A^{-1} \][/tex]
### Step 5: Transpose to return to the [tex]\( X \)[/tex] form
To return to the original form of [tex]\( X \)[/tex], we need to transpose both sides of the equation:
[tex]\[ X^t - I = \left( B^{-1}(A^t - B) A^{-1} \right)^t \][/tex]
### Step 6: Solve for [tex]\( X^t \)[/tex]
Next, we need to isolate [tex]\( X^t \)[/tex]:
[tex]\[ X^t = \left( B^{-1}(A^t - B) A^{-1} \right)^t + I \][/tex]
### Step 7: Transpose to solve for [tex]\( X \)[/tex]
Finally, we transpose the equation to solve for [tex]\( X \)[/tex]:
[tex]\[ X = \left( \left( B^{-1}(A^t - B) A^{-1} \right)^t + I \right)^t \][/tex]
### Final Solution
Hence, the solution for [tex]\( X \)[/tex] is:
[tex]\[ X = \left( \left( B^{-1}(A^t - B) A^{-1} \right)^t + I \right)^t \][/tex]
[tex]\[ \left(B\left(X^t - I\right)^t A\right)^t + B = A^t \][/tex]
We will solve this step-by-step.
### Step 1: Transpose the entire equation
First, we need to eliminate the outer transpose by transposing the entire equation:
[tex]\[ \left( \left(B\left(X^t - I\right)^t A\right)^t \right)^t + B^t = \left(A^t\right)^t \][/tex]
Since transposing twice returns the original matrix, we have:
[tex]\[ B\left(X^t - I\right)^t A + B^t = A \][/tex]
### Step 2: Rearrange the term involving [tex]\( B \)[/tex]
We need to simplify the equation:
[tex]\[ B\left(X^t - I\right)^t A + B = A^t \][/tex]
### Step 3: Subtract [tex]\( B \)[/tex] from both sides
Move [tex]\( B \)[/tex] to the right-hand side of the equation:
[tex]\[ B\left(X^t - I\right)^t A = A^t - B \][/tex]
### Step 4: Isolate the term involving [tex]\( X \)[/tex]
Next, we need to deal with the term [tex]\(\left(X^t - I\right)^t\)[/tex]. As [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are non-singular, they have inverses [tex]\( A^{-1} \)[/tex] and [tex]\( B^{-1} \)[/tex]. We will use these inverses to isolate the term [tex]\(\left(X^t - I\right)^t\)[/tex]:
Multiply both sides by [tex]\( B^{-1} \)[/tex] from the left:
[tex]\[ \left(X^t - I\right)^t A = B^{-1}(A^t - B) \][/tex]
Next, multiply both sides by [tex]\( A^{-1} \)[/tex] from the right:
[tex]\[ \left(X^t - I\right)^t = B^{-1}(A^t - B) A^{-1} \][/tex]
### Step 5: Transpose to return to the [tex]\( X \)[/tex] form
To return to the original form of [tex]\( X \)[/tex], we need to transpose both sides of the equation:
[tex]\[ X^t - I = \left( B^{-1}(A^t - B) A^{-1} \right)^t \][/tex]
### Step 6: Solve for [tex]\( X^t \)[/tex]
Next, we need to isolate [tex]\( X^t \)[/tex]:
[tex]\[ X^t = \left( B^{-1}(A^t - B) A^{-1} \right)^t + I \][/tex]
### Step 7: Transpose to solve for [tex]\( X \)[/tex]
Finally, we transpose the equation to solve for [tex]\( X \)[/tex]:
[tex]\[ X = \left( \left( B^{-1}(A^t - B) A^{-1} \right)^t + I \right)^t \][/tex]
### Final Solution
Hence, the solution for [tex]\( X \)[/tex] is:
[tex]\[ X = \left( \left( B^{-1}(A^t - B) A^{-1} \right)^t + I \right)^t \][/tex]
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