How do we know the ith of an invertible matrix B is orthogonal to the jth column of B^-1 , if i is not equal/unequal to j?
First please confirm the following typographical correction indicated in bold:
"How do we know the ith row of an invertible matrix B is orthogonal to the jth column of B^-1 , if i is not equal/unequal to j?"
By definition,
BB-1=I
which by matrix multiplication, the inner product of the ith row of B and the jth column of B-1 results in the element Iij, or the element on the ith row and the jth column.
Since the unit matrix I has the property that Iii=1 and Iij=0 if i≠j, the question is answered, namely the ith row of B and the jth column of B-1 are orthogonal.
Well, considering invertible matrices and their properties, it's like having a good punchline to a joke. When "i" and "j" are not equal, their relationship is more like a comedy duo rather than an orthogonal pair. They don't really have that nice, perpendicular dynamic going on. So, unfortunately, there's no guarantee that the "ith" of an invertible matrix is orthogonal to the "jth" column of its inverse. It's a bit like trying to find humor in a serious mathematical situation - sometimes the punchline just doesn't land.
To determine whether the i-th row of an invertible matrix B is orthogonal to the j-th column of B^-1 (where i is not equal to j), we need to examine the properties of orthogonal matrices.
An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors. This means that the dot product, or inner product, of any two different columns or rows is zero.
Given that B is an invertible matrix, we can state that B^-1 also exists. To determine if the i-th row of B is orthogonal to the j-th column of B^-1, we need to compute their dot product.
Let's denote the i-th row of B as r_i and the j-th column of B^-1 as c_j^-1.
The dot product r_i . c_j^-1 of these vectors is given by:
r_i . c_j^-1 = r_i * c_j^-1
For the dot product to be zero (indicating orthogonality), the result of this multiplication must be zero.
Therefore, if the dot product r_i . c_j^-1 equals zero, we can conclude that the i-th row of B is orthogonal to the j-th column of B^-1.
To determine if the ith row of an invertible matrix B is orthogonal to the jth column of B^(-1), where i is not equal to j, we can use the property of the dot product.
1. First, consider the definition of the dot product between two vectors. The dot product of two vectors u and v is given by the sum of the products of their corresponding entries:
u · v = u₁v₁ + u₂v₂ + ... + uₙvₙ
2. With this in mind, let's assume that B is an invertible matrix and let's denote the ith row of B as Bᵢ and the jth column of B^(-1) as (B^(-1))ⱼ.
3. To check if Bᵢ is orthogonal to (B^(-1))ⱼ, we can compute their dot product (Bᵢ · (B^(-1))ⱼ) and check if it equals zero.
4. However, directly calculating Bᵢ · (B^(-1))ⱼ can be time-consuming, since it involves multiplying the entries of the ith row of B with the entries of the jth column of B^(-1).
5. An alternative approach is to use the fact that the dot product of two vectors is zero if and only if they are orthogonal.
6. Therefore, instead of computing the dot product directly, we can check if the ith row of B⋅B^(-1) is orthogonal to the jth column of the identity matrix.
7. Recall that B⋅B^(-1) equals the identity matrix, denoted by I.
8. To verify if the ith row of I is orthogonal to the jth column of I, we can compute their dot product (Iᵢ · Iⱼ) and check if it equals zero.
9. Since the diagonal elements of the identity matrix are all equal to 1, except for the entry at the intersection of row i and column j of I, which is 0, their dot product will be equal to 0 if i is not equal to j.
10. Hence, if i is not equal to j, we can conclude that the ith row of B is orthogonal to the jth column of B^(-1).
By following these steps, you can determine if the ith row of an invertible matrix B is orthogonal to the jth column of B^(-1), where i is not equal to j.