A non-singular matrix is invertible, has independent columns and rows, and has a unique solution to equations of the form Ax=b. A singular matrix is not invertible, has dependent columns and rows, and the equation Ax=b may have no solution or multiple solutions. Key differences between singular and non-singular matrices include whether the determinant is zero, the number of solutions to Ax=0, the matrix rank, and whether eigenvalues can be zero.

1. Singular and Non-singular Matrix
Yuji Oyamada
1 HVRL, Keio University
2 Chair for Computer Aided Medical Procedure (CAMP), Technische Universit¨t M¨nchen
a u
April 10, 2012

2. Non-Singular
A is non-singular means that A is invertible (A−1 exists).
Can solve Ax = b as ˆ = A−1 b.
x
The above solution is unique.
For homogeneous system Ax = 0, the only solution is x = 0.
Y. Oyamada (Keio Univ. and TUM) Singular and Non-singular Matrix April 10, 2012 2/4

3. Singular
A is singular means that A is not invertible (A−1 doet not exist).
Either
a solution to Ax = b does not exist,
there is more than one solution (not unique).
The homogeneous system Ax = 0 has more than one solution.
Infinitely many non-trivial solutions.
Y. Oyamada (Keio Univ. and TUM) Singular and Non-singular Matrix April 10, 2012 3/4

4. Comparison
Non-singular Singular
A is invertible not invertible
Columns independent dependent
Rows independent dependent
det(A) =0 =0
Ax = 0 one solution x = 0 infinitely many solution
Ax = b one solution no solution or infinitely many
A has n (nonzero) pivots r < n pivots
A has full rank r = n rank r < n
Column space is all of Rn has dimension r < n
Row space is all of Rn has dimension r < n
Eigenvalue All eigenvalues are non-zero Zero is an eigenvalue of A
AT A is symmetric positive definite is only semidefinite
Singular value of A has n (positive) singular values has r < n singular values
Y. Oyamada (Keio Univ. and TUM) Singular and Non-singular Matrix April 10, 2012 4/4