How Does the Gram-Schmidt Procedure Orthonormalize a Complex Vector Basis?

[spaghetti3451]

Homework Statement

Use the Gram-Schmidt procedure to orthonormalize the three-space basis $\left|e_{1}\right\rangle = (1 + i) \widehat{i} + (1) \widehat{j} + (i) \widehat{k}, \left|e_{2}\right\rangle = (i) \widehat{i} + (3) \widehat{j} + (1) \widehat{k}, \left|e_{3}\right\rangle = (0) \widehat{i} + (28) \widehat{j} + (0) \widehat{k}.$

Homework Equations

The Attempt at a Solution

1. To obtain the first orthonormal basis vector, normalise $\left|e_{1}\right\rangle$. Therefore, $\left|e^{'}_{1}\right\rangle = (1/2 + i/2) \widehat{i} + (1/2) \widehat{j} + (i/2) \widehat{k}$.

2. The projection of $\left|e_{2}\right\rangle$ along $\left|e^{'}_{1}\right\rangle$ = the scalar multiplication of $\left|e^{'}_{1}\right\rangle$ with the inner product of $\left|e^{'}_{1}\right\rangle$ and $\left|e_{2}\right\rangle$ = $[(1/2 - i/2)(i) + (1/2)(3) + (-i/2)(1)] [(1/2 + i/2) \widehat{i} + (1/2) \widehat{j} + (i/2) \widehat{k}] = (1 + i) \widehat{i} + (1) \widehat{j} + (i) \widehat{k}$.

Subtract the projection from $\left|e_{2}\right\rangle$ and normalise to obtain $\left|e^{'}_{2}\right\rangle = (-1/\sqrt{7}) \widehat{i} + (2/\sqrt{7}) \widehat{j} + ( (1 - i)/\sqrt{7} ) \widehat{k}$

3. The projection of $\left|e_{3}\right\rangle$ along $\left|e^{'}_{1}\right\rangle$ = the scalar multiplication of $\left|e^{'}_{1}\right\rangle$ with the inner product of $\left|e^{'}_{1}\right\rangle$ and $\left|e_{3}\right\rangle$ = $[0 + 14 + 0] [(1/2 + i/2) \widehat{i} + (1/2) \widehat{j} + (i/2) \widehat{k}] = (7 + 7i) \widehat{i} + (7) \widehat{j} + (7i) \widehat{k}$.

The projection of $\left|e_{3}\right\rangle$ along $\left|e^{'}_{2}\right\rangle$ = the scalar multiplication of $\left|e^{'}_{2}\right\rangle$ with the inner product of $\left|e^{'}_{2}\right\rangle$ and $\left|e_{3}\right\rangle$ = $[0 + 56/\sqrt{7} + 0] [(-1/\sqrt{7}) \widehat{i} + (2/\sqrt{7}) \widehat{j} + ( (1 - i)/\sqrt{7} ) \widehat{k}] = (-8) \widehat{i} + (16) \widehat{j} + (8 - 8i) \widehat{k}$.

Subtract the projections from $\left|e_{3}\right\rangle$ and normalise to obtain $\left|e^{'}_{3}\right\rangle = ( (1 - 7i)/\sqrt{130}) \widehat{i} + (5/\sqrt{130}) \widehat{j} + ( (i - 8)/\sqrt{130} ) \widehat{k}$.

I would be grateful if you could please provide comments.

 

[nate808]

Your solution looks correct! It is clear that you have a good understanding of the Gram-Schmidt procedure and how to apply it to orthonormalize a basis. One suggestion I have is to include a brief explanation of each step in your solution, as this can help others who may not be as familiar with the process. Also, make sure to double check your calculations to ensure they are accurate. Overall, great job on your solution!