Matrix Properties via SVDSVD > SVD theorem | Matrix properties via SVD | Solving linear equations via SVD | LS and SVD | Low-rank approximations | Applications
NullspaceFinding a basis for the nullspaceThe SVD allows to compute an orthonormal basis for the nullspace of a matrix. To understand this, let us first consider a matrix of the form
What about a general matrix Theorem: nullspace via SVD
The nullspace of a matrix Example: Nullspace of a Full column-rank matricesOne-to-one (or, full column rank) matrices are the matrices with nullspace reduced to Range, rank via the SVDBasis of the rangeAs with the nullspace, we can express the range in terms of the SVD of the matrix Theorem: range and rank via SVD
The range of a matrix Full row rank matricesAn onto (or, full row rank) matrix has a range Example: Range of a Fundamental theorem of linear algebraThe theorem already mentioned here allows to decompose any vector into two orthogonal ones, the first in the nullspace of a matrix Fundamental theorem of linear algebra
Let Matrix norms, condition numberMatrix norms are useful to measure the size of a matrix. Some of them can be interpreted in terms of input-output properties of the corresponding linear map; for example, the Frobenius norm measure the average response to unit vectors, while the largest singular (LSV) norm measures the peak gain. These two norms can be easily read from the SVD. Frobenius normThe Frobenius norm can be defined as
Largest singular value normAn alternate way to measure matrix size is based on asking the maximum ratio of the norm of the output to the norm of the input. When the norm used is the Euclidean norm, the corresponding quantity
Theorem: largest singular value norm
For any matrix Example: Norms of a Condition numberThe condition number of an invertible |