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- https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/08%3A_The_Eigenvalue_Problem/8.05%3A_The_Eigenvalue_Problem-_Examples\[\begin{array}{ccc} {P_{1} = e_{1}(e_{1}^{T}e_{1})^{-1}e_{1}^{T}}&{and}&{P_{2} = e_{2}(e_{2}^{T}e_{2})^{-1}e_{2}^{T}} \end{array} \nonumber\] It is not the square root of the sum of squares of its co...\[\begin{array}{ccc} {P_{1} = e_{1}(e_{1}^{T}e_{1})^{-1}e_{1}^{T}}&{and}&{P_{2} = e_{2}(e_{2}^{T}e_{2})^{-1}e_{2}^{T}} \end{array} \nonumber\] It is not the square root of the sum of squares of its components but rather the square root of the sum of squares of the magnitudes of its components. \[\begin{array}{ccc} {P_{1} = e_{1}(e_{1}^{H}e_{1})^{-1}e_{1}^{H}}&{and}&{P_{2} = e_{2}(e_{2}^{H}e_{2})^{-1}e_{2}^{H}} \end{array} \nonumber\]
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/08%3A_The_Eigenvalue_Problem/8.03%3A_The_Partial_Fraction_Expansion_of_the_Resolvent\[R_{j,k+1} R_{j,l+1} = \frac{1}{(2\pi i)^2} \int R(z)(z-\lambda_{j})^{k} dz \int R(w)(w-\lambda_{j})^{l} dw \nonumber\] \[R_{j,k+1} R_{j,l+1} = \frac{1}{(2\pi i)^2} \int R(z) (z-\lambda_{j})^{k} \int...\[R_{j,k+1} R_{j,l+1} = \frac{1}{(2\pi i)^2} \int R(z)(z-\lambda_{j})^{k} dz \int R(w)(w-\lambda_{j})^{l} dw \nonumber\] \[R_{j,k+1} R_{j,l+1} = \frac{1}{(2\pi i)^2} \int R(z) (z-\lambda_{j})^{k} \int \frac{(w-\lambda_{j})^{l}}{w-z} dw dz-\frac{1}{(2\pi i)^2} \int R(w) (w-\lambda_{j})^{k} \int \frac{(z-\lambda_{j})^{k}}{w-z} dz dw \nonumber\] \[D_{j}^{m_{j}} = R_{j, m_{j}+1} = \frac{1}{2\pi i} \int R(z)(z-\lambda_{j})^{m_{j}} dz = 0 \nonumber\]
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/00%3A_Front_Matter/04%3A_PrefaceOur goal in these notes is to demonstrate the role of matrices in the modeling of physical systems and the power of matrix theory in the analysis and synthesis of such systems. In short, the vector of...Our goal in these notes is to demonstrate the role of matrices in the modeling of physical systems and the power of matrix theory in the analysis and synthesis of such systems. In short, the vector of currents is a linear transformation of the vector of voltage drops which is itself a linear transformation of the vector of potentials.
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/05%3A_Differentiation_and_Antidifferentiation/5.07%3A_The_Total_Variation_(Length)_of_a_Function_f_-_E1__E/5.7.E%3A_Problems_on_Total_Variation_and_Graph_Length(i) For \(I=[a, b](a<b), f(x)=\left\{\begin{array}{ll}{1} & {\text { if } x \in R(\text { rational }), \text { and }} \\ {0} & {\text { if } x \in E^{1}-R.}\end{array}\right.\) Then prove that \(V_{h}...(i) For \(I=[a, b](a<b), f(x)=\left\{\begin{array}{ll}{1} & {\text { if } x \in R(\text { rational }), \text { and }} \\ {0} & {\text { if } x \in E^{1}-R.}\end{array}\right.\) Then prove that \(V_{h}[J] \leq V_{f}[I],\) taking an arbitrary \(P^{\prime}=\left\{c=s_{0}, \ldots, s_{m}=d\right\}\), \(\left.\text { and defining } P=\left\{t_{0}, \ldots, t_{m}\right\}, \text { with } t_{i}=g\left(s_{i}\right) . \text { What if } g(c)=b, g(d)=a ?\right]\)
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/04%3A_Function_Limits_and_Continuity/4.03%3A_Operations_on_Limits._Rational_Functions/4.3.E%3A_Problems_on_Continuity_of_Vector-Valued_Functions\left(\forall x \in B \cap G_{\neg p}\left(\delta^{\prime}\right)\right) \quad|f(x)-q|<\frac{\varepsilon}{2} \text { and }\left(\forall x \in B \cap G_{\neg p}\left(\delta^{\prime \prime}\right)\right...\left(\forall x \in B \cap G_{\neg p}\left(\delta^{\prime}\right)\right) \quad|f(x)-q|<\frac{\varepsilon}{2} \text { and }\left(\forall x \in B \cap G_{\neg p}\left(\delta^{\prime \prime}\right)\right) \quad|g(x)-r|<\frac{\varepsilon}{2} .
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/09%3A_Calculus_Using_Lebesgue_Theory/9.04%3A_Convergence_of_Parametrized_Integrals_and_Functions/9.4.E%3A_Problems_on_Uniform_Convergence_of_Functions_and_C-Integrals\left.\int_{0}^{v} t^{-u} \sin t d t \geq \frac{1}{2} \int_{0}^{v} \frac{d t}{t^{u-1}} \rightarrow \infty .\right] \begin{aligned}(\forall x \in A)(\forall \varepsilon>0)\left(\forall y_{0} \in B\righ...\left.\int_{0}^{v} t^{-u} \sin t d t \geq \frac{1}{2} \int_{0}^{v} \frac{d t}{t^{u-1}} \rightarrow \infty .\right] \begin{aligned}(\forall x \in A)(\forall \varepsilon>0)\left(\forall y_{0} \in B\right)(&\exists \delta>0)\left(\forall y \in B \cap G_{y_{0}}(\delta)\right) \\ &\left|F(y)-F\left(y_{0}\right)\right| \leq \int_{A}\left|f(x, y)-f\left(x, y_{0}\right)\right| d m(x) \leq \int_{A}\left(\frac{\varepsilon}{m A}\right) d m=\varepsilon . \end{aligned}
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/02%3A_Matrix_Methods_for_Mechanical_Systems
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/05%3A_Matrix_Methods_for_Dynamical_Systems/5.06%3A_Supplemental_-_Matrix_Analysis_of_the_Branched_Dendrite_Nerve_Fiber\[A^{T}GA = \begin{pmatrix} {G_{i}+G_{cb}}&{-G_{i}}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\ {-G_{i}}&{2G_{i}+G_{m}}&{-G_{i}}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\ {0}&{-G_{i}}&{2G_{i}+G_{m}}&{-G_{i}}&{0}&{0}&{0}&{0...\[A^{T}GA = \begin{pmatrix} {G_{i}+G_{cb}}&{-G_{i}}&{0}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\ {-G_{i}}&{2G_{i}+G_{m}}&{-G_{i}}&{0}&{0}&{0}&{0}&{0}&{0}&{0}\\ {0}&{-G_{i}}&{2G_{i}+G_{m}}&{-G_{i}}&{0}&{0}&{0}&{0}&{0}&{0}\\ {0}&{0}&{-G_{i}}&{3G_{i}}&{-G_{i}}&{0}&{0}&{-G_{i}}&{0}&{0}\\ {0}&{0}&{0}&{-G_{i}}&{2G_{i}+G_{m}}&{-G_{i}}&{0}&{0}&{0}&{0}\\ {0}&{0}&{0}&{0}&{-G_{i}}&{2G_{i}+G_{m}}&{-G_{i}}&{0}&{0}&{0}\\ {0}&{0}&{0}&{0}&{0}&{-G_{i}}&{G_{i}+G_{m}}&{0}&{0}&{0}\\ {0}&{0}&{0}&{-G_{i}}&{0}&{0}&{0}&{2G_{i}+G…
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/03%3A_Vector_Spaces_and_Metric_Spaces/3.04%3A_Complex_Numbers/3.4.E%3A_Problems_on_Complex_Numbers_(Exercises)\begin{aligned} z &=r(\cos \theta+i \sin \theta) \\ z^{\prime} &=r^{\prime}\left(\cos \theta^{\prime}+i \sin \theta^{\prime}\right), \text { and } \\ z^{\prime \prime} &=r^{\prime \prime}\left(\cos \t...\begin{aligned} z &=r(\cos \theta+i \sin \theta) \\ z^{\prime} &=r^{\prime}\left(\cos \theta^{\prime}+i \sin \theta^{\prime}\right), \text { and } \\ z^{\prime \prime} &=r^{\prime \prime}\left(\cos \theta^{\prime \prime}+i \sin \theta^{\prime \prime}\right) \end{aligned}
- https://math.libretexts.org/Bookshelves/Analysis/A_Primer_of_Real_Analysis_(Sloughter)/zz%3A_Back_Matter/20%3A_GlossaryExample and Directions Words (or words that have the same definition) The definition is case sensitive (Optional) Image to display with the definition [Not displayed in Glossary, only in pop-up on pag...Example and Directions Words (or words that have the same definition) The definition is case sensitive (Optional) Image to display with the definition [Not displayed in Glossary, only in pop-up on pages] (Optional) Caption for Image (Optional) External or Internal Link (Optional) Source for Definition "Genetic, Hereditary, DNA ...") (Eg. "Relating to genes or heredity") The infamous double helix CC-BY-SA; Delmar Larsen Glossary Entries Definition Image Sample Word 1 Sample Definition 1
- https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/07%3A_Complex_Analysis_II
