MAT 2250 Definitions, Theorems, etc.

linear equation

A __________ in the variables x₁, x₂, …, x_n has the form a₁x₁ + a₂x₂ + … + a_nx_n = b₁ where the coefficients and b₁ are real numbers. A ___________ either has:

i) exactly one solution

ii) infinitely many solutions

iii) no solution

nonlinear

An equation that is not linear is said to be ______.

system of linear equations (linear system)

A ____________________ is a collection of one or more linear systems involving the same variables, say x₁, x₂, x₃, …, x_n.

solution of the system

A _____________in n variables is a set of n numbers S₁, S₂, …, S_n so that it satisfies each equation when we set x₁ = S₁, x₂ = S₂, …, x_n = S_n.

solution set

The set of all solutions of a linear system is called the _________.

equivalent

Two linear systems are ________ if they have the same solution set.

consistent

A linear system is _______ if it has either 1 solution or infinitely many solutions. A system is in-_______ if it has no solution.

elementary row operations

i) Replacement: replacing a row by the sum of itself with a multiple of another row

ii) Interchange: swapping two rows

iii) Scaling: multiplying all entries of a row by a nonzero constant

row equivalent

Two matrices are ____________ if there exists a sequence of elementary row operations that transforms one system into the other.

• If two augmented matrices are __________, the two systems have the same solution set.

Echelon form (row Echelon form)

A matrix is in __________ if it has the following properties:

i) All nonzero rows are above any rows of all zeros.

ii) Each leading entry of a row is in a column to the right of the leading entry of the row above it.

iii) All entries in a column below a leading entry are zero.

**Using elementary row operations, we can ALWAYS get a matriz in this form

reduced Echelon form (reduced row Echelon form)

If a matrix in Echelon form satisfies the additional conditions, then it is what we call ______________.

iv) The leading entry in each nonzero row is 1.

v) Each leading 1 is the only nonzero entry in its column.

**Using elementary row operations, we can ALWAYS get a matrix in this form

row equivalent echelon matrices

Theorem: Every matrix is row equivalent to one and only one (a unique) reduced Echelon matrix

pivot position

A ___________ in a matrix A is the location in A that corresponds to a leading 1 in the reduced Echelon form of A.

pivot column

A ___________ is a column of a matrix A that contains a pivot position.

row reduction algorithm

Forward phases:

Step 1: Begin with the left-most nonzero column (pivot column). The pivot position is at the top.

Step 2: Select a nonzero entry in the pivot column as a pivot. If necessary, interchange rows to move this entry into the pivot position.

Step 3: Use row replacement operations to create zeros in all positions below the pivot.

Step 4: Cover/ignore the row containing the pivot position and cover all rows, if any, above it. Apply Steps 1-3 to the remaining sub matrix. Repeat the process until there are no more rows to modify.

Backward phase:

Step 5: Begin with the right-most pivot and working upward and to the left, create zeros above each pivot. If a pivot (leading entry) is not 1, make it 1 using scaling.

basic variables and free variables

The variables that correspond to pivot columns are called _____________. The remaining variables, if any, are called ____________

existence and uniqueness

Theorem:

• A linear system is consistent if and only if the right-most column of the augmented matrix is NOT a pivot column. That is, if and only if, an echelon form of the augmented matrix has no row of the form [0 0 … 0 b], where b ≠ 0.

• If a linear system is consistent, then the solution set contains either:

  • a unique solution, where there are no free variables, or

  • infinitely many solutions where there are at least one free variable.

vector equivalence

Two vectors in ℝⁿ are equal if and only if the corresponding entries are equal. Vectors in ℝⁿ are n x 1 matrices or n-tuples with real numbers as entries.

zero vector

The _________ is a vector with only zeros as entries.

Algebraic Properties of ℝⁿ

For all u, v, and w in ℝⁿ and all scalars c and d:

i) Commutative Property: u + v = v + u

ii) Associative Property: (u + v) + w = u + (v + w)

iii) Additive Identity: u + 0 = 0 + u = u

iv) u + (-u) = 0, where -u is the Additive Inverse of u

v) Distributive Property: c(u + v) = cu + cv, where the left-hand side requires n additions and n multiplications, and the right-hand side requires n additions and 2n multiplications.

linear combination

Given the set of vectors v₁, …, v_p in ℝⁿ and given the scalars c₁, …, c_p, the vector y is given by:

y = c₁v₁ + c₂v₂ + … + c_pv_p is called a ____________ of v₁, v₂, …, v_p with weights (coefficients) c₁, c₂, …, c_p.

spanned

If v₁, …, v_p are in ℝⁿ , then the set of all linear combinations of v₁, …, v_p denoted by span {v₁, …, v_p} is the subset of ℝⁿ ________(or generated) by v₁, …, v_p. So span {v₁, …, v_p} is the set of all vectors of the form c₁v₁ + c₂v₂ + … + c_pv_p with c₁, …, c_p as scalars.

Ax = b

Let A be an m x n matrix with columns a₁, a₂, …, a_n. Let x be in ℝⁿ. Then, the product of A and x denoted by Ax is the linear combination of the columns of A using the entries of x as weights. Ax = [a₁ a₂ … a_n] [x_{1 …} x_n] = x₁a₁ + x₂a₂ + … + x_na_n

The number of entries in x is equal to the number of columns of A.

vector form = augmented matrix form

Theorem: If A is an m x n matrix with columns a₁, a₂, …, a_n and if b is in ℝ^m , the matrix equation Ax = b has the same solution set as the vector equation x₁a₁ + x₂a₂ + … + x_na_n = b which in turn has the same solution set as the linear equations whose augmented matrix is [a₁ a₂ … a_n b].

Ax= b equivalence statements

Theorem: Let A be an m x n matrix. Then the following statements are equivalent. That is, for a particular A, either all statements are true or all false.

i) For each b in ℝ^m , the equation Ax = b has a solution.

ii) Each b in ℝ^m is a linear combination of the columns of A.

iii) The columns of A span ℝ^m .

dot product (inner product)

Supposed x = [x₁ … x_n] (image as a column), y = [y₁ … y_m]. The product of y and x is given by yx = [y₁ … y_n][x₁ … x_n] = x₁y₁ + x₂y₂ + … + x_ny_n. yx is called the __________ of y and x.

identity matrix

The _____________ I has 1’s on the diagonal and zero’s elsewhere. If it is n x n, we denote it by I_n. If x is in ℝⁿ , then I_nx = X.

Property of matrix-vector product

Theorem: If A is an m x n matrix, v and u are vectors (in ℝⁿ), c is a scalar, then:

i) A(u + v) = Au + Av (distributive)

ii) A(cu) = c(Au)

homogeneous linear systems

A system of linear equations is said to be _______________ if it can be written in the form Ax = 0. Here, A is an m x n matrix, x is an n x 1 vector, and 0 is an n x 1 zero vector.

The ___________ Ax = 0 ALWAYS has the trivial solution. (x = 0 is always a solution). The ___________ equation Ax = 0 has at least a nontrivial solution if and only if the solution set has at least 1 free variable.

nonhomogeneous systems

A linear system of equations of the form Ax = b with b ≠ 0 is called a _______________.

????

Theorem: Suppose the equation Ax = b is consistent for a given b, and let b be a solution. Then, the solution set of Ax = b is the set of all vectors of the form w = p + v_h, where v_h is any solution of the homogenous equation Ax = 0.

Steps to write the solution of a consistent system of equations in parametric vector form

i) Obtain the reduced row echelon form of the augmented matrix

ii) Express each basic variable in terms of any free variables

iii) Write a typical solution x as a vector whose entries depend on the free variables (if any)

iv) Decompose x into a linear combination of vectors using the free variables as parameters.

linear independence

Consider an indexed set of vectors {v₁, v₂, …, vp} in ℝⁿ. The set of vectors is said to be ________________ if the equation:

c₁v₁ + c₂v₂ + … + c_pv_p = 0 (linear combination) has only the trivial solution c₁ = c₂ = … = c_p = 0.

• A set of TWO nonzero vectors {v₁, v₂} is __________ if and only if neither of the vectors are a multiple of the other

*Also applies to vector spaces

linear dependence

The set of vectors {v₁, v₂, …, vp} is _____________ if there exists weights c₁, c₂, …, c_p not all zero such that:

c₁v₁ + c₂v₂ + … + c_pv_p = 0. That is, the vector equation has at least one nontrivial solution.

• A set of TWO vectors {v₁, v₂} are __________ if one vector is a multiple of the other.

zero vector and linear dependence

Theorem: If a set S = {v₁, v₂, …, v_p} in ℝⁿ contains the zero vector, then the set is linearly dependent

Proof: Let S = {v₁, v₂, …, v_p} be in ℝⁿ. Suppose v₁ = 0. Consider c₁v₁ + … + c_pv_p = 0.

(7)0 + (0)v₂ + … (0)v_p = 0. So c₁ = 7, c₂ = c₃ = … = c_p = 0 is a nontrivial solution. Therefore, the vectors in S are linearly dependent.

linear dependence for sets with more vectors than entries in each vector

Theorem: If a set contains more vectors than there are entries in the vectors, then the set is linearly dependent. That is, any set that has {v₁ … v_p} in ℝⁿ is linearly dependent if p > n.

Proof: Let A = [v₁ … v_p]. Then, A is n x p. Consider the homogeneous equation Ax = 0 (always consistent). Assume p > n. The system has more variables than equations. The system must have free variables. Thus, the system will have nonzero solutions. Thus, the vectors/columns of A {v₁ … v_p} are linearly dependent.

linear dependence of vectors by linear combinations

Theorem: The set S = {v₁ … v_p} of two or more vectors is linearly dependent if and only if at least one vector in S is a linear combination of the other vectors in S.

transformation (mapping or function)

A ____________ T from ℝⁿ to ℝ^m is a rule that assigns to each vector x in ℝⁿ a vector T(x) in ℝ^m. ℝⁿ is called the domain, and ℝ^m is called the codomain of T. (T: ℝⁿ → ℝ^m)

image

For each x in ℝⁿ, the vector T(x) in ℝ^m is called the ______ of x under the action of T. The set of all ________ T(x) is called the range of T. Sometimes codomain = range, but range is typically a subset.

linear transformations

A transformation T is linear if:

i) T(u + v) = T(u) + T(v) for all u and v in the domain of T.

ii) T(cu) = cT(u) for all u in the domain of T and all scalars c.

iii) T(0) = 0

iv) T(cu + dv) = cT(u) + dT(v) for all u and v in the domain of T and all scalars c and d

*So, every matrix transformation is a linear transformation

**Statements i and ii apply to vector spaces

matrix of linear transformations

Theorem: Let T: ℝⁿ → ℝ^m be a linear transformation. Then, there exists a unique matrix A such that T(x) = Ax for all ℝⁿ. A is such that the j^th column is T(e_j). where e_j is the j^th column of I_n. So, A = [T(e₁) T(e₂) … T(e_n)]. A is called the standard matrix for T.

onto

A mapping T: ℝⁿ → ℝ^m is ______ ℝ^m if each b in ℝ^m is the image of at least one x in ℝⁿ.

one-to-one

A mapping T: ℝⁿ → ℝ^m is _______________ if each b in ℝ^m is the image of AT MOST one x in ℝⁿ.

one-to-one transformations with only a trivial solution

Theorem: Let T: ℝⁿ → ℝ^m be a linear transformation. Then, T is 1-1 if and only if the equation T(x) = 0 has only the trivial solution.

onto and 1-1 theorems

Theorem: Let T: ℝⁿ → ℝ^m be a linear transformation and let A be the standard matrix for T. Then:

i) T maps ℝⁿ onto ℝ^m if and only if the columns of A span ℝ^m.

ii) T is 1-1 if and only if the columns of A are linearly independent.

properties of matrix addition

Theorem: Let A, B, and C be matrices of the same size, and let r and s be scalars.

i) A + B = B + A; Commutative

ii) (A + B) + C = A + (B + C); Associative

iii) A + 0 = A; Additive identity

iv) r(A + B) = rA + rB

v) (r + s)A = rA + sA

vi) r(sA) = (rs)A

properties of matrix multiplication

Theorem: Let A be m x n, and B and C with sizes so that the indicated operations can be done. Let r be a scalar.

i) A(BC) = (AB)C; Associative

ii) A (B + C) = AB + AC; Distributive (left)

iii) (B + C)A = BA + BC; Distributive (right)

iv) r(AB) = (rA)B = A(rB)

v) I_nA = A = AI_n

transpose

the __________ of a matrix A is the matrix found by writing its columns as rows (or rows as columns). The __________of A is denoted by A^T.

properties of the transpose

Theorem: Let A and B be matrices whose sizes are appropriate for the following operations:

i) (A^T)^T = A

ii) (A + B)^T = A^T + B^T

iii) (AB)^T = B^TA^T

iv) For any scalar r, (rA)^T = r(A^T)

invertible

An m x n matrix A is said to be _______ if there exists an n x m matrix B called the inverse of A such that BA = I and AB = I. A matrix that is _______ is sometimes called “non-singular”. A matrix that is NOT _________ is called “singular”.

uniqueness of the inverse

The inverse of a matrix A is unique.

formula for the inverse for a two by two matrix

A^-1 = (1/ad - bc)[d -b -c a], where (ad - bc) ≠ 0.

finding x in Ax = b using inverse properties

Theorem: If A is an m x n invertible matrix, then for each b in ℝⁿ the equation Ax = b has the unique solution x = A^-1b

properties of the inverse of matrices

i) If A is invertible, then A^-1 is invertible, and (A^-1)^-1 = A.

ii) If A and B are order n (n x n) invertible matrices, then so is AB, and the inverse of AB is given by (AB)^-1 = B^-1A^-1. Supposed A and B are n x n invertible matrices. Then, A^-1 and B^-1 exist. (AB)(B^-1A^-1) - A(BB^-1)A^-1 = AIA^-1 = (AA^-1)I = I.

iii) If A is invertible, then A^T is invertible. The inverse of A^T is (A^T)^-1 = (A^-1)^T.

iv) Theorem: A square matrix is invertible if det(A) ≠ 0

elementary matrix

An m x n matrix is called an _________________ if it can be obtained from the identity I_n by applying a single elementary row operation.

• If we pre-multiply on the left of matrix A by an ________________, we get a matrix that is equal to the matrix we get when we apply the same operation on A.

• Elementary matrices are invertible, and the inverse of an _________________ is an ____________________ of the same type.

characterization of the invertible matrix

Let A be a square n x n matrix. Then, the following statements are equivalent. That is, for a given A, the statements are either all true or all false.

i) A is an invertible matrix.

ii) A is row equivalent to the n x n identity matrix.

iii) A has n pivot positions

iv) The equation Ax = 0 has only the trivial solution.

v) The columns of A form a linearly independent set.

vi) The linear transformation x → Ax is one-to-one.

vii) The equation Ax = b has at least one solution for each b in ℝⁿ.

viii) The columns of A span ℝⁿ

ix) The linear transformation x → Ax maps ℝⁿ onto ℝⁿ.

x) There is an n x n matrix C such that CA = I.

xi) There is an n x n matrix D such that AD = I

xii) A^T is an invertible matrix.

xiii) The columns of A form a basis for ℝⁿ

xiv) col(A) = ℝⁿ

xv) dim(col(A)) = n

xvi) rank(A) = n

xvii) Null(A)= {0}

xviii) nullity(A) = dim(Null(A)) = 0

xix) The number 0 is NOT an eigenvalue of A

xx) The det(A) ≠ 0

^{*if 0 was an eigenvalue, the determinant would be zero}

invertibility of linear transformations pt. 1

Theorem: A linear transformation T: ℝⁿ → ℝ^m is said to be invertible if there exists a function S: ℝⁿ → ℝⁿ such that

S(T(x)) = x for all x in ℝⁿ (i)

T(S(x)) = x for all x in ℝⁿ (ii)

invertibility of linear transformations pt. 2

Theorem: Let T: ℝⁿ → ℝⁿ be a linear transformation. Let A be the standard matrix for T. Then T is invertible if and only if A is an invertible matrix. In that case, the linear transformation S given by S(x) = A^-1x is the unique function satisfying equations (i) and (ii)

cofactor

Supposed A is a square matrix. A_ij is the matrix obtained from A by deleting the i^th row and the j^th column of A. The _________ is given by c_ij = (-1)^i+j | A_ij |.

determinant of an n x n matrix

If A is an n x n matrix, then the determinant of A is the sum of the entries of the first row of A times their cofactors. That is det(A) = a₁₁c₁₁ + a₁₂c₁₂ + … a_1nc_1n = summation of a_1kc_1k where k = 1 and goes to n = summation of a_1k(-1)^1+kdet(A_1k) where k = 1 and goes to n.

expansion by cofactors

Theorem: Let A be an n x n matrix, then the determinant of A is given by…

Expansion across i^th row: | A | = a_i1c_i1 + … + a_inc_in

Expansion down j^th column: | A | = a_1jc_1j + … + a_njc_nj

triangular matrix

a matrix where all the entries above or below the main diagonal are zeros

main diagonal

the diagonal of a matrix where the row index equals the column index

upper triangular

A matrix is ___________ if a_ij = 0 for i > j (AKA the space below the main diagonal are all zero entries)

lower triangular

A matrix is ____________ if a_ij = 0 for i < j (AKA the space above the main diagonal are all zero entries)

diagonal

A matrix is _________ if a_ij = 0 for i ≠ j.

determinant of a triangular matrix

Theorem: If A is a triangular matrix, then the determinant of A is the product of the elements along the main diagonal.

properties of determinants

Theorem: Let A be a square matrix.

i) If a multiple of 1 row of A is added to another row to produce B, then det(B) = det(A)

• replacement operations do not change the determinant

ii) If 2 rows of A are interchanged to produce B, then det(B) = -det(A)

iii) If 1 row of A is multiplied by α, then det(B) = αdet(A)

determinants of n x n transposed matrices

Theorem: If A is n x n, then det(A^T) = det(A)

determinant of multiplied matrices

Theorem: If A and B are n x n matrices, then the det(AB) = det(A)det(B)

• using this theorem also proves that det(ABC) = det(A)det(B)det(C) where C is also an n x n matrix

vector space

A _______________ V is a non-empty set of objects, called vectors, on which are defined two operations called addition and multiplication by scalars, subject to ten axioms listed below. The axioms must hold for all vectors u, v, and w in V and scalars c and d.

Addition:

i) The sum of u and v denoted by u + v is in V (Closure under Addition)

ii) u + v = v + u (Commutative property)

iii) (u + v) + w = u + (v + w) (Associative property)

iv) There is a zero vector 0 such that u + 0 = u (Additive identity)

v) For each u in V, there is a vector -u (Additive inverse) in V such that u + (-u) = 0

Multiplication:

vi) The scalar multiple of u by c denoted by cu is in V (Closure under scalar multiplication)

vii) c(u + v) = cu + cv (Distributive property)

viii) (c + d)u = cu + du (Distributive property)

ix) c(du) = cd(u) (Associative property)

x) 1u = u (Scalar identity)

subspace

A _________ W of a vector space V is a subset W of V that has the 3 properties:

i) The zero vector is in W.

ii) W is closed under addition. So if u and v are in W, then u + v is in W.

iii) W is closed under multiplication by a scalar. If u is in W and c is a scalar, then cu is in W.

**Different than subset

symmetric

Matrix A is __________ if A^T = A.

skew-symmetric

Matrix A is _______________ if A^T = -A

spanning set (generated set)

Given a subspace H of a vector space V, a _____________ for H is a set of vectors S = {v₁, …, v_p} such that H = span(S)

The Null Space of A (kernel)

N(A) = {x: x is in ℝⁿ and Ax = 0}.

Supposed A is an m x n matrix. Let N(A) be the set of all solutions to the homogeneous equation Ax = 0

row space of A

The ____________________ is the subspace of ℝⁿ spanned by the row vectors of A.

column space of A

The ____________________ is the subspace of ℝⁿ spanned by the columns of A. The __________________ is denoted by col(A).

range (in vector spaces)

The _____ of a linear transformation T is the set of all vectors in the vector space W of the form T(x).

____(T) = {y in W: T(x) = y, x is in V}

basis

Let H be a subspace of a vector space V. A set of vectors B = { v₁, …, v_p} is a _____ for H if

i) B is a linearly independent set

ii) H = span(B)

**If H is V, then B is a _____ for V

Spanning Set Theorem

Theorem: Let S = {v₁, …, v_p} be a set in V and H = span(S)

i) If one of the vectors in S, say V_k, is a linear combination of the remaining vectors in S, then the set formed from S by removing V_k still spans H.

iI) If H ≠ {0}, some subset of S is a basis for H.

basis and column spaces

Theorem: The columns of a matrix A that correspond to the pivot columns of A form a basis for col(A).

Uniqueness of Representation

Theorem: Let B = {b₁, .., b_n} be a basis for a vector space V. Then, for each x in V, there exists a unique set of scalars c₁, c₂, …, c_n such that x = c₁b₁ + c₂b₂ + … + c_nb_n}

Proof:

Supposed B = {b₁, …, b_n} is a basis for a vector space V and x is an element in V. Since B spans V, then there are scalars c₁, …, c_n such that x = c₁b₁ + … + c_nb_n. Now assume there is another way to represent x in terms of B. That is x = d₁b₁ + … + d_nb_n for some scalars d₁, …, d_n.

Subtract the two equations from each other and get

0 = (c₁b₁ + … + c_nb_n) - (d₁b₁ + … + d_nb_n)

0 = (c₁ - d₁)b₁ + … + (c_n - d_n)b_n.

In the final equation, we have the linear combination of the vectors in B equal to zero. Since B is linearly independent, then all the coefficients must be zero. Thus, the representation is unique.

the coordinates of x relative to the basis of B (or B-coordinates of x)

Supposed B = {b₁, …, b_n} is a basis for a vector space V. The ________________________________________ are the unique weights such that x = c₁b₁ + … + c_nb_n

If c₁, …, c_n are the ____________________, then the vector [x]_B = [c₁ … c_n] in ℝⁿ is the _________________.

The mapping x → [x]_B is the ______________________________ determined by B

standard basis

The _____________ is the simplest, most conventional basis for a vector space, consisting of orthogonal unit vectors that define the coordinate axes.

transition matrix from one coordinate system to another

P[x]_B = [x]_S where P is the matrix called the change of coordinates matrix, or the ___________________

The coordinate mapping

Supposed V is a vector space (maybe different from ℝⁿ) with a basis B = {b₁, …, b_n}. The basis introduces a coordinate system in ℝⁿ. For any x in V, we have c₁, …, c_n such that x = c₁b₁ + … + c_nb_n

[x]_B = [c₁ … c_n]. So we have a mapping that takes x → [x]_B that connects a possibly unfamiliar vector space V to a familiar space ℝ_n

coordinate mapping as a linear transformation

Theorem: Let B = {b₁, …, b_n} be a basis for a vector space V. Then, the coordinate mapping x → [x]_B is a one-to-one linear transformation from V onto ℝⁿ.

In general, a one-to-one linear transformation from a vector space V onto a vector space W is called an isomorphism.

checking linear dependence with vectors in a vector space given dimension of a basis

Theorem: If a vector space V has basis B = {b₁, …, b_n}, then any set in V containing more than n vectors must be linearly dependent.

basis’s in vector spaces and amount of vectors

Theorem: If a vector space V has a basis of n vectors, then every basis of V must consist of exactly n vectors.

finite-dimensional

If V is spanned by a finite set, then V is set to be ___________________, and the dimension of V written as dim(V) is the number of vectors in a basis of V. If V is NOT spanned by a finite number of vectors, then V is set to be in__________________.

expanding of a linearly independent subset to become a basis

Theorem: Let H be a subspace of a finite-dimensional vector space V. Any linearly independent set can be expanded, if necessary, to be a basis of H. Also, H is finite-dimensional and dim(H) <= dim(V).

qualifications for being a basis if dim(set) = dim(vector space) and is linearly indp/spans

Theorem: Let V be a p-dimensional vector space, p >= 1. Any linearly independent set of exactly p vectors in V is automatically a basis for V. OR Any set of exactly p vectors that spans V is automatically a basis for V.

**Note: The dimension of Null(A) is the number of free variables in the equation Ax = 0

**Note: The dimension of col(A) is the number of pivot columns of A.

row equivalence and equivalent basis’s of two matrices

Theorem: If two matrices A and B are row equivalent, then their row spaces are the same. If B is in row-echelon form, the nonzero rows of B form a basis for the row space of A, as well as B.

rank

The ____ of A is the dimension of col(A).

nullity

The _______ of A is the dimension of Null(A).

Rank Theorem

Theorem: The dimension of the column space and the row space of an m x n matrix A are equal. This common dimensions, the rank of A, also equals the number of pivot columns of A and satisfies the equation rank(A) + nullity(A) = n AKA the # of cols in A

eigenvalues and eigenvectors

Suppose A is an n x n matrix. We want to find all values of lambda and NONZERO x such that Ax = (lambda)x. Any pair (lambda)x that satisfies the equation is called an eigenpair of A, where (lambda) is called the ____________ of A and x is the corresponding _____________.

characteristic equation

det(A - lambda(I)) = 0

characteristic polynomial

a(lambda)² + b(lambda) + c = 0

where a, b, and c are scalars and lambda is the unknown variable