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Whether you're a super-fan of matrix decompositions, or a desperate linear algebra student, Omni's polar decomposition calculator will be your faithful servant. We'll discuss the methods of computing polar decomposition and go together through the following questions:

  • What can be said about the existence of polar decomposition?
  • Is polar decomposition unique?

💡 This tool is part of our matrix decomposition collection. Other calculators in this category are:

Make sure to visit one of them once you're done here!

What is the polar decomposition of a matrix?

Let A be a matrix of size n×n. There exists two matrices, U and P — both of size n×n — such that A = UP. Furthermore,

  • If A is real, then U is orthogonal and P is a positive semi-definite symmetric matrix.
  • If A is complex, then U is unitary and P is a positive semi-definite Hermitian matrix.

It could help to view the polar decomposition as the equivalent of the polar form of a complex number for matrices. With polar form, a complex number z can be written as z = r × exp(iθ). Here, r is a non-negative value and exp(iθ) comes down to a complex number with unit length. r and exp(iθ) correspond to P and U from the polar decomposition, respectively.

We can determine P as P = √(A*A). Since A*A is a positive-semidefinite Hermitian matrix, it has a unique positive-semidefinite Hermitian square root.

Next, if A is invertible, then U can be found from the equation A = UP by multiplying it from the right by the inverse of P. If A is non-invertible, then it's not that easy to find U. A more efficient approach to computing polar decomposition is via the singular value decomposition (SVD).

How do I find the polar decomposition?

To quickly and efficiently find the polar decomposition of a matrix A using the singular value decomposition:

  1. Write down the SVD decomposition of A. Assume A = WSV*.
  2. Compute P = VSV*.
  3. Compute U = WV*.
  4. These are your matrices P and U! You can verify directly that A = UP.

As you can see, computing polar decomposition is not that trivial: we have to perform SVD, and then lots more time-consuming matrix multiplications. Fortunately, our polar decomposition calculator will do that for you!

How to use this polar decomposition calculator?

Our polar decomposition calculator is pretty straightforward to use. You need to:

  1. Pick the matrix A's size.
  2. Enter the matrix coefficients in their respective fields.
  3. The polar decomposition of your matrix will appear at the bottom of our tool.
  4. By default, the results have the precision of 4 decimal places. You can adjust this by clicking the Advanced mode button underneath this polar decomposition calculator.

Does the polar decomposition always exist?

Yes, every matrix has a polar decomposition. Since polar decomposition can be found via singular value decomposition, the existence of polar decomposition follows from the existence of the SVD.

Is polar decomposition unique?

In general, no. Polar decomposition is unique for invertible matrices. However, if the matrix is singular (non-invertible), then its polar decomposition is not unique.

Anna Szczepanek, PhD
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