Help with Linear Algebra exercise

[vslo]

Homework Statement

Hi guys,
I am new to this forum. I got a final exam tomorrow and the professor told us to solve some exercise before it. I came up with one exercise that I don't know how to do, at all.
Hope you guys can give me some light. Here it goes.

Know that the multiplication of a matrix by a vector can be write as :

$$\left[ \begin{array}{c} b1 \\ . \\ bn \end{array} \right]\; =\; \left[ \begin{array}{ccc} A11 & . & A1n \\ . & . & . \\ Am1 & . & Amn \end{array} \right]\; .\; \left[ \begin{array}{c} x1 \\ . \\ xn \end{array} \right]$$

or like $$bi=\sum_{j=1}^{n}{\; } \mbox{Ai}jxj\; ,\; i\; =1,2,3,...,m\; ,\; j=\; 1,2,3,\; ...,\; n$$


a) prove that :
||$$\vec{d}$$ - $$\vec{w}$$ $$\underline{x}$$||² = $$\sum_{i=1}^{m}{\; }\left( d_{i}\; -x_{i}^{T}w\; \; \right)^{2}$$

where $$\vec{d}$$= [d₁ d₂ d₃...d_m]^T

W$$\in$$Rⁿ

$$\vec{x}$$_i=[x_i1 x_i2 ... x_in ]^T

$$\underline{x}$$= [$$\vec{x}$$₁ $$\vec{x}$$₂ ... $$\vec{x}$$_m ]

b) Prove that $$\underline{x}$$^T$$\underline{x}$$ is real and simetric.

Obs: $$\underline{x}$$ means the matrix x

Homework Equations

The Attempt at a Solution

None of my attempts were close to something

Hope you guys understand the question and give me a hand !

 

[vslo]

anyone, Please?

 

[hunt_mat]

If x is a (1,n) row matrix, then x^T*x will be a (n,n) matrix, I presume that it is this that your lecturer wanted to prove that this matrix is symmetric?
The way I would go about it is this:
1) Write down the definition of the product of two matrices
2) Specialise to the case which you're interested (i.e. one row vector and column vector)
3) Compute the transpose of this matrix.
4) Compare with the original matrix to see if the elements are the same.

I can't help you with the first part as you have either misquoted it and not told me the norm you're using.

Mat

 

[Susanne217]

vslo are you sure you copied question one correctly?

 

[Mark44]

The b vector in this equation -
$$\left[ \begin{array}{c} b1 \\ . \\ bn \end{array} \right]\; =\; \left[ \begin{array}{ccc} A11 & . & A1n \\ . & . & . \\ Am1 & . & Amn \end{array} \right]\; .\; \left[ \begin{array}{c} x1 \\ . \\ xn \end{array} \right]$$

should have m entries, not n.

 

[vslo]

The b vector in this equation -
$$\left[ \begin{array}{c} b1 \\ . \\ bn \end{array} \right]\; =\; \left[ \begin{array}{ccc} A11 & . & A1n \\ . & . & . \\ Am1 & . & Amn \end{array} \right]\; .\; \left[ \begin{array}{c} x1 \\ . \\ xn \end{array} \right]$$

should have m entries, not n.

Yes, he is right. the b vector has M entries, not N... Does it helps ?
Thank you for your observation Mark !