Expressing a quadratic form with a matrix | Summary and Q&A

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June 16, 2016
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Khan Academy
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Expressing a quadratic form with a matrix

TL;DR

The video explains how to express quadratic forms in a vectorized sense using matrices and vectors, allowing for easy scalability and generalization.

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Questions & Answers

Q: What is a quadratic form?

A quadratic form is an expression that consists only of quadratic terms, where variables are multiplied by constants and added together. It does not include linear terms or constants outside the quadratic terms.

Q: How can quadratic forms be expressed in a vectorized form?

Quadratic forms can be expressed in a vectorized form by using matrices and vectors. A matrix, usually a symmetric 2x2 matrix, is multiplied with the variable vector and its transpose.

Q: Why is it more convenient to write quadratic forms in a vectorized form?

Writing quadratic forms in a vectorized form allows for easy scalability and generalization. The notation remains the same even when dealing with larger matrices and vectors, which is useful when working with higher dimensions or a larger number of variables.

Q: How is the multiplication of a matrix with a vector done in a quadratic form?

When multiplying a matrix with a vector in a quadratic form, each corresponding term in the matrix's rows is multiplied with the corresponding term in the vector. This results in a new vector with the computed terms.

Summary & Key Takeaways

  • Quadratic forms are expressions that consist solely of quadratic terms.

  • By using matrices and vectors, quadratic forms can be expressed in a vectorized form, similar to linear terms.

  • The vectorized form of a quadratic form involves multiplying a matrix with the variable vector and its transpose.

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