Equivalent Conditions for Singular and Nonsingular Matrices
Let A be an n × n matrix. Any pair of statements in the same column are equivalent.
A is singular (A−1 does not exist). A is nonsingular (A−1 exists).
Rank(A) = n. Rank(A) = n.
|A| = 0. |A| = 0.
A is not row equivalent to In . A is row equivalent to In .
AX = O has a nontrivial solution for X. AX = O has only the trivial solution
for X.
AX = B does not have a unique AX = B has a unique solution
solution (no solutions or for X (namely, X = A−1 B).
infinitely many solutions).
The rows of A do not form a The rows of A form a
basis for Rn . basis for Rn .
The columns of A do not form The columns of A form
a basis for Rn . a basis for Rn .
The linear operator L: Rn → Rn The linear operator L: Rn → Rn
given by L(X) = AX is given by L(X) = AX
not an isomorphism. is an isomorphism.
Diagonalization Method
To diagonalize (if possible) an n × n matrix A:
Step 1: Calculate pA (x) = |xIn − A|.
Step 2: Find all real roots of pA (x) (that is, all real solutions to pA (x) = 0). These are the
eigenvalues λ1 , λ2 , λ3 , . . . , λk for A.
Step 3: For each eigenvalue λm in turn:
Row reduce the augmented matrix [λm In − A | 0] . Use the result to obtain a
set of particular solutions of the homogeneous system (λm In − A)X = 0 by setting
each independent variable in turn equal to 1 and all other independent variables
equal to 0.
Step 4: If after repeating Step 3 for each eigenvalue, you have less than n fundamental
eigenvectors overall for A, then A cannot be diagonalized. Stop.
Step 5: Otherwise, form a matrix P whose columns are these n fundamental eigenvectors.
Step 6: Verify that D = P−1 AP is a diagonal matrix whose dii entry is the eigenvalue for the
fundamental eigenvector forming the ith column of P. Also note that A = PDP−1 .
,Simplified Span Method (Simplifying Span(S))
Suppose that S is a finite subset of Rn containing k vectors, with k ≥ 2.
To find a simplified form for span(S), perform the following steps:
Step 1: Form a k × n matrix A by using the vectors in S as the rows of A. (Thus, span(S) is
the row space of A.)
Step 2: Let C be the reduced row echelon form matrix for A.
Step 3: Then, a simplified form for span(S) is given by the set of all linear combinations of
the nonzero rows of C.
Independence Test Method (Testing for Linear Independence of S)
Let S be a finite nonempty set of vectors in Rn .
To determine whether S is linearly independent, perform the following steps:
Step 1: Create the matrix A whose columns are the vectors in S.
Step 2: Find B, the reduced row echelon form of A.
Step 3: If there is a pivot in every column of B, then S is linearly independent. Otherwise,
S is linearly dependent.
Coordinatization Method (Coordinatizing v with Respect to an Ordered Basis B)
Let V be a nontrivial subspace of Rn , let B = (v1, . . . , vk ) be an ordered basis for V,
and let v ∈ Rn . To calculate [v]B , if it exists, perform the following:
Step 1: Form an augmented matrix [A | v] by using the vectors in B as the columns of A,
in order, and using v as a column on the right.
Step 2: Row reduce [A | v] to obtain the reduced row echelon form [C | w].
Step 3: If there is a row of [C | w] that contains all zeroes on the left and has a nonzero
entry on the right, then v ∈ / span(B) = V, and coordinatization is not possible.
Stop.
Step 4: Otherwise, v ∈ span(B) = V. Eliminate all rows consisting entirely of zeroes in [C | w]
to obtain [Ik | y]. Then, [v]B = y, the last column of [Ik | y].
Transition Matrix Method (Calculating a Transition Matrix from B to C )
To find the transition matrix P from B to C where B and C are ordered bases for a
nontrivial k-dimensional subspace of Rn , use row reduction on
⎡ ⎤
1st 2nd kth 1st 2nd kth
⎢ vector vector · · · vector vector vector · · · vector ⎥
⎣ in in in in in in ⎦
C C C B B B
to produce
Ik P
.
rows of zeroes
,Elementary Linear
Algebra
, Elementary Linear
Algebra
Fifth Edition
Stephen Andrilli
Department of Mathematics
and Computer Science
La Salle University
Philadelphia, PA
David Hecker
Department of Mathematics
Saint Joseph’s University
Philadelphia, PA
AMSTERDAM • BOSTON • HEIDELBERG • LONDON
NEW YORK • OXFORD • PARIS • SAN DIEGO
SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
Academic Press is an imprint of Elsevier
Let A be an n × n matrix. Any pair of statements in the same column are equivalent.
A is singular (A−1 does not exist). A is nonsingular (A−1 exists).
Rank(A) = n. Rank(A) = n.
|A| = 0. |A| = 0.
A is not row equivalent to In . A is row equivalent to In .
AX = O has a nontrivial solution for X. AX = O has only the trivial solution
for X.
AX = B does not have a unique AX = B has a unique solution
solution (no solutions or for X (namely, X = A−1 B).
infinitely many solutions).
The rows of A do not form a The rows of A form a
basis for Rn . basis for Rn .
The columns of A do not form The columns of A form
a basis for Rn . a basis for Rn .
The linear operator L: Rn → Rn The linear operator L: Rn → Rn
given by L(X) = AX is given by L(X) = AX
not an isomorphism. is an isomorphism.
Diagonalization Method
To diagonalize (if possible) an n × n matrix A:
Step 1: Calculate pA (x) = |xIn − A|.
Step 2: Find all real roots of pA (x) (that is, all real solutions to pA (x) = 0). These are the
eigenvalues λ1 , λ2 , λ3 , . . . , λk for A.
Step 3: For each eigenvalue λm in turn:
Row reduce the augmented matrix [λm In − A | 0] . Use the result to obtain a
set of particular solutions of the homogeneous system (λm In − A)X = 0 by setting
each independent variable in turn equal to 1 and all other independent variables
equal to 0.
Step 4: If after repeating Step 3 for each eigenvalue, you have less than n fundamental
eigenvectors overall for A, then A cannot be diagonalized. Stop.
Step 5: Otherwise, form a matrix P whose columns are these n fundamental eigenvectors.
Step 6: Verify that D = P−1 AP is a diagonal matrix whose dii entry is the eigenvalue for the
fundamental eigenvector forming the ith column of P. Also note that A = PDP−1 .
,Simplified Span Method (Simplifying Span(S))
Suppose that S is a finite subset of Rn containing k vectors, with k ≥ 2.
To find a simplified form for span(S), perform the following steps:
Step 1: Form a k × n matrix A by using the vectors in S as the rows of A. (Thus, span(S) is
the row space of A.)
Step 2: Let C be the reduced row echelon form matrix for A.
Step 3: Then, a simplified form for span(S) is given by the set of all linear combinations of
the nonzero rows of C.
Independence Test Method (Testing for Linear Independence of S)
Let S be a finite nonempty set of vectors in Rn .
To determine whether S is linearly independent, perform the following steps:
Step 1: Create the matrix A whose columns are the vectors in S.
Step 2: Find B, the reduced row echelon form of A.
Step 3: If there is a pivot in every column of B, then S is linearly independent. Otherwise,
S is linearly dependent.
Coordinatization Method (Coordinatizing v with Respect to an Ordered Basis B)
Let V be a nontrivial subspace of Rn , let B = (v1, . . . , vk ) be an ordered basis for V,
and let v ∈ Rn . To calculate [v]B , if it exists, perform the following:
Step 1: Form an augmented matrix [A | v] by using the vectors in B as the columns of A,
in order, and using v as a column on the right.
Step 2: Row reduce [A | v] to obtain the reduced row echelon form [C | w].
Step 3: If there is a row of [C | w] that contains all zeroes on the left and has a nonzero
entry on the right, then v ∈ / span(B) = V, and coordinatization is not possible.
Stop.
Step 4: Otherwise, v ∈ span(B) = V. Eliminate all rows consisting entirely of zeroes in [C | w]
to obtain [Ik | y]. Then, [v]B = y, the last column of [Ik | y].
Transition Matrix Method (Calculating a Transition Matrix from B to C )
To find the transition matrix P from B to C where B and C are ordered bases for a
nontrivial k-dimensional subspace of Rn , use row reduction on
⎡ ⎤
1st 2nd kth 1st 2nd kth
⎢ vector vector · · · vector vector vector · · · vector ⎥
⎣ in in in in in in ⎦
C C C B B B
to produce
Ik P
.
rows of zeroes
,Elementary Linear
Algebra
, Elementary Linear
Algebra
Fifth Edition
Stephen Andrilli
Department of Mathematics
and Computer Science
La Salle University
Philadelphia, PA
David Hecker
Department of Mathematics
Saint Joseph’s University
Philadelphia, PA
AMSTERDAM • BOSTON • HEIDELBERG • LONDON
NEW YORK • OXFORD • PARIS • SAN DIEGO
SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO
Academic Press is an imprint of Elsevier
