A matrix P \in \mathbb {R}^{d \times d} is orthogonal (sometimes referred to as a rotation matrix ) if P P^ T = P^ T P = I_ d. Suppose that
P = \begin{pmatrix} v_1 & v_2 & \cdots & v_ d \\ \end{pmatrix}
where v_1, v_2, \ldots , v_ d \in \mathbb {R}^ d are column vectors.
Is the identity matrix I_ d an orthogonal matrix?
Yes
No
unanswered
What is \sum _{i = 1}^ d (v_1^ i)^2?
unanswered
What is v_1 \cdot v_2?
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Are the rows of P unit vectors?
Yes
No
Yes
Yes, the identity matrix I_d is an orthogonal matrix.
The sum of (v_1^i)^2, where i ranges from 1 to d, is equal to 1. This is because the columns of an orthogonal matrix are orthonormal, meaning the dot product of each column vector with itself is equal to 1.
The dot product of v_1 and v_2 can be calculated as v_1 · v_2 = v_1^T v_2, where ^T represents the transpose of a vector. Without the specific values of v_1 and v_2, it is not possible to determine the exact value of their dot product.
The rows of matrix P are not necessarily unit vectors. The definition of an orthogonal matrix states that the columns of P must be orthonormal (i.e., unit vectors), but it does not require the rows to be unit vectors.
To determine if the identity matrix I_d is orthogonal, we need to check if I_d satisfies the condition P P^T = P^T P = I_d.
Now, P is given as a matrix with column vectors v_1, v_2, ..., v_d. The product P P^T represents the matrix multiplication of P with its transpose, and P^T P represents the matrix multiplication of the transpose of P with P.
So, to find P P^T, we need to multiply the matrix P with its transpose. However, since P is the identity matrix I_d, its transpose is also equal to I_d.
Hence, P P^T = I_d * I_d = I_d.
Similarly, to find P^T P, we need to multiply the transpose of P with P. Again, since P is the identity matrix I_d, its transpose is also equal to I_d.
Hence, P^T P = I_d * I_d = I_d.
Therefore, I_d satisfies the condition P P^T = P^T P = I_d, which means the identity matrix I_d is an orthogonal matrix.
Now, to answer the remaining questions:
1. The expression \sum_{i=1}^d (v_1^i)^2 represents the sum of the squares of the components of the vector v_1. This can be calculated by squaring each component of v_1 and summing them together.
2. The expression v_1 \cdot v_2 represents the dot product of the vectors v_1 and v_2. This can be calculated by multiplying the corresponding components of v_1 and v_2 and summing them together.
3. The rows of matrix P are unit vectors if each row vector has a magnitude of 1. To check this, you can calculate the magnitude of each row vector of P by taking the square root of the sum of the squares of its components. If the magnitude is equal to 1, then the row vector is a unit vector.