施密特正交化（Gram-Schmidt method to orthonormalize vectors）

施密特正交化算法

Give an arbitrary basis for an n-dimensional inner product space V, we can constructs an orthogonal basis for V.

the Gram-Schmidt algorithm is:

Step 1: Let ；

Step 2: Let ；

Step 3: Let

.

.

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计算举例

Let with the Euclidean inner product. We will apply the Gram-Schmidt algorithm to orthogonalize the basis .

Step 1

Let , so .

Step 2

Let .

.

Step 3

Let .

You can verify that forms an orthogonal basis for . Normalizing the vectors in the orthogonal basis, we obtain the orthonormal basis

.