What is the transpose of a vector?

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.

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 .

Which is the product of a vector and its transpose?

In the derivation that you cite, the vectors a and b are being treated as n × 1 matrices, so a T is a 1 × n matrix. 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:

Which is the transpose and dot product of an M nmatrix?

Transpose & Dot Product Def: The transpose of an m nmatrix Ais the n mmatrix ATwhose columns are the rows of A. So: The columns of ATare the rows of A. The rows of ATare the columns of A. Example: If A= \ 1 2 3 4 5 6 \ , then AT= 2 4 1 4 2 5 3 6 3 5: Convention: From now on, vectors v 2Rnwill be regarded as \\columns” (i.e.: n 1 matrices).

How to transpose a vector to a RN vector?

In summary, A: Rn -> Rm, At: Rm -> Rn, x is in Rn, y is in Rm. x |-> x’ = A*x, which is in Rm. y |-> y’ = At*y, which is in Rn. < x’, y > = < x, y’ >, and so whether we send x to y’s home (Rm) via A, or send y to x’s home (Rn) via At, we end up with two vectors, which are “equally aligned” in either case. Hope ths helps!

Is the matrix multiplied with its transpose something special?

If attention is restricted to real-valued (non-singular square invertible) matrices, then an appropriate question and some answers are found in Polar decomposition of real matrices . Especially the following formula over there leaves no doubt that a matrix multiplied with its transpose IS something special: