What happens when a vector is multiplied by its transpose?

What happens when a vector is multiplied by its transpose?

If A is an m × n matrix and AT is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: A AT is m × m and AT A is n × n. Furthermore, these products are symmetric matrices. Similarly, the product AT A is a symmetric matrix.

What happens when you multiply a matrix by a vector?

So, if A is an m×n matrix, then the product Ax is defined for n×1 column vectors x . If we let Ax=b , then b is an m×1 column vector. In other words, the number of rows in A determines the number of rows in the product b .

Is a transpose a always symmetric?

The product of any matrix (square or rectangular) and it’s transpose is always symmetric.

Can a vector have a transpose?

The transpose of a vector is vT ∈R1×m a matrix with a single row, known as a row vector. A special case of a matrix-matrix product occurs when the two factors correspond to a row multiplying a column vector.

Is a 1×3 matrix a vector?

A point or a vector is a sequence of three numbers and for this reason they too can be written as a 1×3 matrix, a matrix that has one row and three columns: Point written in a matrix form P=[xyz].

What does a matrix do to a vector?

We define the matrix-vector product only for the case when the number of columns in A equals the number of rows in x. So, if A is an m×n matrix (i.e., with n columns), then the product Ax is defined for n×1 column vectors x. If we let Ax=b, then b is an m×1 column vector.

What do you mean by transpose matrix?

The transpose of a matrix is simply a flipped version of the original matrix. We can transpose a matrix by switching its rows with its columns. We denote the transpose of matrix A by AT. For example, if A=[123456] then the transpose of A is AT=[142536].

Which is an example of a transpose of a matrix?

The transpose of a matrix is simply a flipped version of the original matrix. We can transpose a matrix by switching its rows with its columns. We denote the transpose of matrix A by A T. For example, if A T = [ 1 4 2 5 3 6]. We can take a transpose of a vector as a special case.

Which is the product of a vector and its transpose?

By the rules of matrix multiplication, a T a and a T b result in a 1 × 1 matrix, which is equivalent to a scalar, while a a T produces an n × n matrix: a a T = [ a 1 a 2 ⋮ a n] [ a 1 a 2 ⋯ a n] = [ a 1 2 a 1 a 2 ⋯ a 1 a n a 2 a 1 a 2 2 ⋯ a 2 a n ⋮ ⋮ ⋱ ⋮ a n a 1 a n a 2 ⋯ a n 2].

When does a matrix hold in a vector space?

If you have a real vector space equipped with a scalar product, and an Orthogonal matrix A then A A T = I holds. A matrix is orthogonal if for the scalar product ⟨ v, w ⟩ = ⟨ A v, A w ⟩ holds for any v, w ∈ V

Is the projection vector p and its transpose the same?

The projection vector p since it lies on a is: To me both a a T and a T a are dot products and the order shouldn’t matter. Then p = b. But it is not. Why? You appear to be conflating the dot product a ⋅ b of two column vectors with the matrix product a T b, which computes the same value.