Term
| Vector Space Properties of Matrices (VSPM) |
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Definition
Suppose that [image] is the set of all mxn matrices with addition and scalar multiplication as defined in definitions MA and MSM. Then if [image]
- ACM Addition Closure
[image]
- SCM Scalar Multiplication Closure
[image]
- CM Commutivity
[image]
- AAM Additive Associtivity
[image]
- ZM Zero Matrix
[image] for all [image]
- AIM Additive Inverse
[image]
- SCAM Scalar Multiplication Associtivity
[image]
- DMAM Distributivity across Addition
[image]
- DSAM Distributivity across Scalar Addition
[image]
- OM One Matrix
[image]
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Term
| Symmetric Matrices are Square (SYM) |
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Definition
Suppose that A is a symmetric matrix. Then A is square. |
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Term
| Transpose and Matrix Addition (TMA) |
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Definition
Suppose that A and B are mxn matrices. Then
[image] |
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Term
| Transpose and Matrix Scalar Multiplication(TMSM) |
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Definition
Suppose that [image] and A is an mxn matrix. Then
[image] |
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Term
| Transpose of a Transpose (TT) |
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Definition
Suppose that A is an mxn matrix. Then
[image] |
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Term
| Conjugation Respects Matrix Addition (CRMA) |
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Definition
Suppose that [image] and A is an mxn matrix. Then
[image] |
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Term
| Conjugation Respects Matrix Scalar Multiplication (CRMSM) |
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Definition
Suppose that [image] and A is an mxn matrix. Then
[image] |
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Term
| Conjugate of a Conjugate of a Matrix (CCM) |
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Definition
Suppose that A is an mxn matrix. Then
[image] |
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Term
| Matrix Conjugation and Transpose (MCT) |
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Definition
Suppose that A is an mxn matrix. Then
[image] |
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Term
| Ajdoint and Matrix Addion (AMA) |
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Definition
Suppose that A and B are matrices of the same size. Then
[image] |
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Term
| Adjoint and Matrix Scalare Multiplication (AMSM) |
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Definition
Suppose that [image] is a scalar and A is a matrix. Then
[image] |
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Term
| Adjoint of an Adjoint (AA) |
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Definition
Suppose that A is a matrix. Then
[image] |
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Term
| Systems of Linear Equations as Matrix Multiplication (SLEMM) |
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Definition
The set of solutions to the linear system LS(A,b) equals the set of solutions for x in the vector equation Ax =b. |
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Term
| Eqaul Matrices and Matrix-Vector Products (EMMVP) |
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Definition
Suppose that A and B are m x n matrices such that Ax = Bx for every [image]. Then A=B. |
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Term
| Entries of Matrix Products (EMP) |
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Definition
Suppose that A is an m x n matrix and B is an n x p matrix. Then for [image], the individual entries of AB are given by
[image]
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Term
| Matrix Multiplication and the Zero Matrix (MMZM) |
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Definition
Suppose that A is an m x n matrix. Then
- [image]
- [image]
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Term
| Matrix Multiplication and Identity Matrix (MMIM) |
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Definition
Suppose tha A is an m x n matrix. Then
- [image]
[image]
[This is why the Identity Matrix has its name; it acts like a scalar one for matrix multiplication.] |
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Term
| Matrix Multiplication Distributes Across Addition (MMDAA) |
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Definition
Suppose A is an m x n matrix and B and C are n x p and n x s matrices and D is a p x n matrix. Then
- A(B + C) = AB + AC
- (B + C)D = BD + CD
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Term
| Matrix Multiplication and Scalar Matrix Multiplication (MMSM) |
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Definition
Suppose that A is an m x n matrix and B is an n x p matrix. Let [image] be a scalar. Then
[image] |
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Term
| Matrix Multiplication is Associative (MMA) |
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Definition
Suppose A is an m x n matrix, B is an n x p matrix and D is an p x s matrix. Then
A(BD) = (AB)D |
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Term
| Matrix Multiplication and Inner Product (MMIP) |
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Definition
If we consider the vectors [image] matrices, then
[image] |
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Term
| Matrix Multiplication and Complex Conjugation (MMCC) |
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Definition
Suppose that A is an m x n matrix and B is an n x p matrix. Then
[image] |
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Term
| Matrix Multiplication and Transpose (MMT) |
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Definition
Suppose A is an m x n matrix and B is an n x p matrix. Then
[image] |
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Term
| Matrix Multiplication and Adjoints (MMAD) |
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Definition
Suppose A is an m x n matrix and B is an n x p matrix.
[image] |
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Term
| Adjoint and Inner Product (AIP) |
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Definition
Suppose that A is an m x n matrix and [image]. Then
[image] |
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Term
| Hermitian Matrices and Inner Products (HMIP) |
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Definition
Suppose that A is a square matrix of size n. Then A is Hermitian if and only if [image].
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Term
| Two-by-Two Matrix Inverse (TTMI) |
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Definition
Suppose [image]. Then A is invertible if and only if [image].
When A is invertible then
[image] |
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Term
| Matrix Inverse is Unique (MIU) |
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Definition
Suppose the square matrix A has an inverse. Then [image] is unique. |
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Term
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Definition
Suppose A and B are invertible matrices of size n. Then AB is an invertible matrix and [image]. |
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Term
| Matrix Inverse of a Matrix Inverse (MIMI) |
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Definition
Suppose A is an invertible matrix. Then [image] is invertible and [image]. |
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Term
| Matrix Inverse of a Transpose (MIT) |
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Definition
Suppose A is an invertible matrix. Then [image] is invertible and [image]. |
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Term
| Matrix Inverse of a Scalar Multiple (MISM) |
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Definition
Suppose A is an invertible matrix and [image] is a non-zero scalar. Then [image] is invertible. |
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Term
| Nonsingular Product has Nonsingular Terms (NPNT) |
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Definition
Suppose that A and B are square matrices of size n. Then product AB is nonsingular if and only if A and B are nonsingular. |
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Term
| One-Sided Inverse is Sufficient (OSIS) |
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Definition
Suppose A and B are square matrices of size n such that [image], then [image]. |
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Term
| Nonsingularity is Invertibility (NI) |
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Definition
Suppose that A is a square matrix. Then A is nonsingular if and only if A is invertible. |
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Term
| Solution with Nonsingular Coefficient Matrix (SNCM) |
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Definition
Suppose that A is nonsingular. Then the unique solution to [image] is [image]. |
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Term
| Unitary Matrices are Invertible (UMI) |
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Definition
Suppose that U is a unitary matrix of size n. Then U is nonsingular and [image]. |
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Term
| Columns of Unitary Matrices are Orthonormal Sets (CUMOS) |
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Definition
Suppose that [image] is the set of columns of a square matrix A of size n. Then A is a unitary matrix if and only if S is an orthonormal set. |
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Term
| Unitary Matrices Preserve Inner Products (UMPIP) |
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Definition
Suppose that U is a unitary matrix of size n and [image] and [image] are vectors from [image]. Then [image] and [image]. |
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Term
| Column Spaces and Consistent Systems (CSCS) |
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Definition
Suppose A is an m x n matrix and b is a vector of size m. Then [image] if and only if LS(A,b) is consistent. |
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Term
| Basis of the Column Space (BCS) |
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Definition
Suppose that A is an m x n matrix with columns [image] and B is a row-equivalent matrix in reduced row-echelon form with r non-zero rows.
Let [image] be the set of indices for the pivot columns of B. Let [image]. Then
- T is a linearly independent set
- [image]
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Term
| Column Space of a Nonsingular Matrix (CSNM) |
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Definition
Suppose A is a square matrix of size n. Then A is nonsingular if and only if [image]. |
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Term
| Row-Equivalent Matrices have equal Row Spaces (REMRS) |
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Definition
Suppose A and B are row-equivalent matrices. Then R(A) = R(B). |
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Term
| Basis for the Row Space (BRS) |
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Definition
Suppose A is a matrix and B is a row-equivalent matrix in reduced row-echelon form. Let S be the set of non-zero columns in [image]. Then
- [image]
- S is a linearly independent set
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Term
| Column Space, Row Space, Transpose (CSRST) |
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Definition
Suppose A is a matrix. Then [image]. |
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