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Hi I'm stuck on the following question and I have little idea as to how to proceed.

Note: I only know how to calculate eigenvalues of a matrix, I don't many applications of them(apart from finding powers of matrices). Also, I will denote the inner product by <a,b> rather than with circular brackets as it done in the question. I will incorporate these changes in the wording of the question.

a) Suppose that <u,v> is an inner product on R^n. Define n by n matrix [tex]A = \left[ {a_{ij} } \right][/tex] by [tex]a_{ij} = \left\langle {e_i ,e_j } \right\rangle [/tex]. Show that, if we regard [tex]\mathop u\limits^ \to = \left( {b_1 ,...,b_n } \right)[/tex] and [tex]\mathop v\limits^ \to = \left( {c_1 ,...,c_n } \right)[/tex] as column vectors, then:

[tex]

\left\langle {u,v} \right\rangle = \mathop u\limits^ \to ^T A\mathop v\limits^ \to = \left[ {b_1 ,..,b_n } \right]\left[ {\begin{array}{*{20}c}

{a_{11} } & \cdots & {a_{1n} } \\

\vdots & \vdots & \vdots \\

{a_{n1} } & \cdots & {a_{nn} } \\

\end{array}} \right]\left[ {\begin{array}{*{20}c}

{c_1 } \\

\vdots \\

{c_n } \\

\end{array}} \right]

[/tex]... equation 1

b) Explain why the matrix A in part (a)is symmetric. Can the eigenvalues of A be complex, negative or zero? Justify your answer.

c) For any two vectors u = (a_1, a_2, a_3) and v = (b_1, b_2, b_3) in R^3, define a function:

[tex]

g\left\langle {u,v} \right\rangle = 4a_1 b_1 + 2a_2 b_2 + 3a_3 b_3 + \sqrt 2 a_2 b_3 + \sqrt 2 a_3 b_2

[/tex]

Determine whether g<u,v> is an inner product on R^3. Justify your answers, either directly or by appealing to the answers of the previous parts (a) and (b(b).

a) I'm thinking that with the way things have been defined in the question, that every entry of A on the

Carrying out the matrix multiplication(the bit with the three matrices):

[tex]

\left[ {b_1 ,..,b_n } \right]\left[ {\begin{array}{*{20}c}

{a_{11} } & \cdots & {a_{1n} } \\

\vdots & \vdots & \vdots \\

{a_{n1} } & \cdots & {a_{nn} } \\

\end{array}} \right]\left[ {\begin{array}{*{20}c}

{c_1 } \\

\vdots \\

{c_n } \\

\end{array}} \right] = \left[ {b_1 ,..,b_n } \right]\left[ {\begin{array}{*{20}c}

{c_1 } \\

\vdots \\

{c_n } \\

\end{array}} \right] = \left[ {b_1 c_1 + ... + b_n c_n } \right]

[/tex]

This is a 1 by 1 matrix so it can be regarded as a real number right? I still don't see how this means that equation 1 is true. I mean the inner product isn't necessarily the dot product.

b) Why is A symmetric? It just is? The identity matrix is symmetric assuming that I haven't gotten definitions mixed up so why would an explanation be needed. I'm thinking I got A interpreted incorrectly. I'm not sure about the eigenvalues, isn't there some relationship between eigenvalues and symmetric matrices? The eigenvalues are the diagonal entries? If A is the identity matrix then the eigenvalue/s is 1?

c) As I indicated earlier I don't know enough about eigenvalues to use them to answer this question. Can someone help me out with this question and the others as well? Any help would be great thanks.

Note: I only know how to calculate eigenvalues of a matrix, I don't many applications of them(apart from finding powers of matrices). Also, I will denote the inner product by <a,b> rather than with circular brackets as it done in the question. I will incorporate these changes in the wording of the question.

a) Suppose that <u,v> is an inner product on R^n. Define n by n matrix [tex]A = \left[ {a_{ij} } \right][/tex] by [tex]a_{ij} = \left\langle {e_i ,e_j } \right\rangle [/tex]. Show that, if we regard [tex]\mathop u\limits^ \to = \left( {b_1 ,...,b_n } \right)[/tex] and [tex]\mathop v\limits^ \to = \left( {c_1 ,...,c_n } \right)[/tex] as column vectors, then:

[tex]

\left\langle {u,v} \right\rangle = \mathop u\limits^ \to ^T A\mathop v\limits^ \to = \left[ {b_1 ,..,b_n } \right]\left[ {\begin{array}{*{20}c}

{a_{11} } & \cdots & {a_{1n} } \\

\vdots & \vdots & \vdots \\

{a_{n1} } & \cdots & {a_{nn} } \\

\end{array}} \right]\left[ {\begin{array}{*{20}c}

{c_1 } \\

\vdots \\

{c_n } \\

\end{array}} \right]

[/tex]... equation 1

b) Explain why the matrix A in part (a)is symmetric. Can the eigenvalues of A be complex, negative or zero? Justify your answer.

c) For any two vectors u = (a_1, a_2, a_3) and v = (b_1, b_2, b_3) in R^3, define a function:

[tex]

g\left\langle {u,v} \right\rangle = 4a_1 b_1 + 2a_2 b_2 + 3a_3 b_3 + \sqrt 2 a_2 b_3 + \sqrt 2 a_3 b_2

[/tex]

Determine whether g<u,v> is an inner product on R^3. Justify your answers, either directly or by appealing to the answers of the previous parts (a) and (b(b).

a) I'm thinking that with the way things have been defined in the question, that every entry of A on the

*off*-diagonal are zero since by definition [tex]i \ne j \Rightarrow \left\langle {e_i ,e_j } \right\rangle = 0[/tex]. It*looks*like A is the indentity matrix.Carrying out the matrix multiplication(the bit with the three matrices):

[tex]

\left[ {b_1 ,..,b_n } \right]\left[ {\begin{array}{*{20}c}

{a_{11} } & \cdots & {a_{1n} } \\

\vdots & \vdots & \vdots \\

{a_{n1} } & \cdots & {a_{nn} } \\

\end{array}} \right]\left[ {\begin{array}{*{20}c}

{c_1 } \\

\vdots \\

{c_n } \\

\end{array}} \right] = \left[ {b_1 ,..,b_n } \right]\left[ {\begin{array}{*{20}c}

{c_1 } \\

\vdots \\

{c_n } \\

\end{array}} \right] = \left[ {b_1 c_1 + ... + b_n c_n } \right]

[/tex]

This is a 1 by 1 matrix so it can be regarded as a real number right? I still don't see how this means that equation 1 is true. I mean the inner product isn't necessarily the dot product.

b) Why is A symmetric? It just is? The identity matrix is symmetric assuming that I haven't gotten definitions mixed up so why would an explanation be needed. I'm thinking I got A interpreted incorrectly. I'm not sure about the eigenvalues, isn't there some relationship between eigenvalues and symmetric matrices? The eigenvalues are the diagonal entries? If A is the identity matrix then the eigenvalue/s is 1?

c) As I indicated earlier I don't know enough about eigenvalues to use them to answer this question. Can someone help me out with this question and the others as well? Any help would be great thanks.

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